3.18.82 \(\int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx\) [1782]

Optimal. Leaf size=280 \[ -\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}} \]

[Out]

-2/3*(-a*d+b*c)^3*(-c*f+d*e)*(f*x+e)^(3/2)/d^5-2/5*(-a*d+b*c)^3*(f*x+e)^(5/2)/d^4+2/7*b*(3*a^2*d^2*f^2-3*a*b*d
*f*(c*f+d*e)+b^2*(c^2*f^2+c*d*e*f+d^2*e^2))*(f*x+e)^(7/2)/d^3/f^3-2/9*b^2*(-3*a*d*f+b*c*f+2*b*d*e)*(f*x+e)^(9/
2)/d^2/f^3+2/11*b^3*(f*x+e)^(11/2)/d/f^3+2*(-a*d+b*c)^3*(-c*f+d*e)^(5/2)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d
*e)^(1/2))/d^(13/2)-2*(-a*d+b*c)^3*(-c*f+d*e)^2*(f*x+e)^(1/2)/d^6

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Rubi [A]
time = 0.18, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 214} \begin {gather*} \frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac {2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}-\frac {2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac {2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]

[Out]

(-2*(b*c - a*d)^3*(d*e - c*f)^2*Sqrt[e + f*x])/d^6 - (2*(b*c - a*d)^3*(d*e - c*f)*(e + f*x)^(3/2))/(3*d^5) - (
2*(b*c - a*d)^3*(e + f*x)^(5/2))/(5*d^4) + (2*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*
f + c^2*f^2))*(e + f*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(9/2))/(9*d^2*f^3) +
(2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt
[d*e - c*f]])/d^(13/2)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx &=\int \left (\frac {b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{d^3 f^2}+\frac {(-b c+a d)^3 (e+f x)^{5/2}}{d^3 (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{d^2 f^2}+\frac {b^3 (e+f x)^{9/2}}{d f^2}\right ) \, dx\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {(b c-a d)^3 \int \frac {(e+f x)^{5/2}}{c+d x} \, dx}{d^3}\\ &=-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)\right ) \int \frac {(e+f x)^{3/2}}{c+d x} \, dx}{d^4}\\ &=-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)^2\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^5}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)^3\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^6}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left (2 (b c-a d)^3 (d e-c f)^3\right ) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^6 f}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.62, size = 435, normalized size = 1.55 \begin {gather*} \frac {2 \sqrt {e+f x} \left (231 a^3 d^3 f^3 \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+99 a^2 b d^2 f^2 \left (-105 c^3 f^3+15 d^3 (e+f x)^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )-33 a b^2 d f \left (-315 c^4 f^4+45 c d^3 f (e+f x)^3+5 d^4 (2 e-7 f x) (e+f x)^3+105 c^3 d f^3 (7 e+f x)-21 c^2 d^2 f^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+b^3 \left (-3465 c^5 f^5+495 c^2 d^3 f^2 (e+f x)^3+55 c d^4 f (2 e-7 f x) (e+f x)^3+1155 c^4 d f^4 (7 e+f x)-231 c^3 d^2 f^3 \left (23 e^2+11 e f x+3 f^2 x^2\right )+5 d^5 (e+f x)^3 \left (8 e^2-28 e f x+63 f^2 x^2\right )\right )\right )}{3465 d^6 f^3}-\frac {2 (-b c+a d)^3 (-d e+c f)^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]

[Out]

(2*Sqrt[e + f*x]*(231*a^3*d^3*f^3*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2*(23*e^2 + 11*e*f*x + 3*f^2*x^2)) + 9
9*a^2*b*d^2*f^2*(-105*c^3*f^3 + 15*d^3*(e + f*x)^3 + 35*c^2*d*f^2*(7*e + f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x +
3*f^2*x^2)) - 33*a*b^2*d*f*(-315*c^4*f^4 + 45*c*d^3*f*(e + f*x)^3 + 5*d^4*(2*e - 7*f*x)*(e + f*x)^3 + 105*c^3
*d*f^3*(7*e + f*x) - 21*c^2*d^2*f^2*(23*e^2 + 11*e*f*x + 3*f^2*x^2)) + b^3*(-3465*c^5*f^5 + 495*c^2*d^3*f^2*(e
+ f*x)^3 + 55*c*d^4*f*(2*e - 7*f*x)*(e + f*x)^3 + 1155*c^4*d*f^4*(7*e + f*x) - 231*c^3*d^2*f^3*(23*e^2 + 11*e
*f*x + 3*f^2*x^2) + 5*d^5*(e + f*x)^3*(8*e^2 - 28*e*f*x + 63*f^2*x^2))))/(3465*d^6*f^3) - (2*(-(b*c) + a*d)^3*
(-(d*e) + c*f)^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(13/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(963\) vs. \(2(248)=496\).
time = 0.14, size = 964, normalized size = 3.44

method	result	size
derivativedivides	\(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {11}{2}} d^{5}}{11}+\frac {\left (\left (a d f -b c f \right ) b^{2} d^{4}+b d \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (a d f -b c f \right ) \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )+b d \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {\left (\left (a d f -b c f \right ) \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )+b d \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )+b d \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right ) \sqrt {f x +e}\right )}{d^{6}}-\frac {2 f^{3} \left (a^{3} c^{3} d^{3} f^{3}-3 a^{3} c^{2} d^{4} e \,f^{2}+3 a^{3} c \,d^{5} e^{2} f -a^{3} d^{6} e^{3}-3 a^{2} b \,c^{4} d^{2} f^{3}+9 a^{2} b \,c^{3} d^{3} e \,f^{2}-9 a^{2} b \,c^{2} d^{4} e^{2} f +3 a^{2} b c \,d^{5} e^{3}+3 a \,b^{2} c^{5} d \,f^{3}-9 a \,b^{2} c^{4} d^{2} e \,f^{2}+9 a \,b^{2} c^{3} d^{3} e^{2} f -3 a \,b^{2} c^{2} d^{4} e^{3}-b^{3} c^{6} f^{3}+3 b^{3} c^{5} d e \,f^{2}-3 b^{3} c^{4} d^{2} e^{2} f +b^{3} c^{3} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{6} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\)	\(964\)
default	\(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {11}{2}} d^{5}}{11}+\frac {\left (\left (a d f -b c f \right ) b^{2} d^{4}+b d \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (a d f -b c f \right ) \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )+b d \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {\left (\left (a d f -b c f \right ) \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )+b d \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )+b d \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right ) \sqrt {f x +e}\right )}{d^{6}}-\frac {2 f^{3} \left (a^{3} c^{3} d^{3} f^{3}-3 a^{3} c^{2} d^{4} e \,f^{2}+3 a^{3} c \,d^{5} e^{2} f -a^{3} d^{6} e^{3}-3 a^{2} b \,c^{4} d^{2} f^{3}+9 a^{2} b \,c^{3} d^{3} e \,f^{2}-9 a^{2} b \,c^{2} d^{4} e^{2} f +3 a^{2} b c \,d^{5} e^{3}+3 a \,b^{2} c^{5} d \,f^{3}-9 a \,b^{2} c^{4} d^{2} e \,f^{2}+9 a \,b^{2} c^{3} d^{3} e^{2} f -3 a \,b^{2} c^{2} d^{4} e^{3}-b^{3} c^{6} f^{3}+3 b^{3} c^{5} d e \,f^{2}-3 b^{3} c^{4} d^{2} e^{2} f +b^{3} c^{3} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{6} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\)	\(964\)
risch	\(\text {Expression too large to display}\)	\(1593\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

2/f^3*(1/d^6*(1/11*b^3*(f*x+e)^(11/2)*d^5+1/9*((a*d*f-b*c*f)*b^2*d^4+b*d*(2*a*b*d^4*f-2*b^2*d^4*e))*(f*x+e)^(9
/2)+1/7*((a*d*f-b*c*f)*(2*a*b*d^4*f-2*b^2*d^4*e)+b*d*(a^2*d^4*f^2-a*b*c*d^3*f^2-a*b*d^4*e*f+b^2*c^2*d^2*f^2-b^
2*c*d^3*e*f+b^2*d^4*e^2))*(f*x+e)^(7/2)+1/5*((a*d*f-b*c*f)*(a^2*d^4*f^2-a*b*c*d^3*f^2-a*b*d^4*e*f+b^2*c^2*d^2*
f^2-b^2*c*d^3*e*f+b^2*d^4*e^2)+b*d*(-a^2*c*d^3*f^3+a^2*d^4*e*f^2+a*b*c^2*d^2*f^3-a*b*d^4*e^2*f-b^2*c^2*d^2*e*f
^2+b^2*c*d^3*e^2*f))*(f*x+e)^(5/2)+1/3*((a*d*f-b*c*f)*(-a^2*c*d^3*f^3+a^2*d^4*e*f^2+a*b*c^2*d^2*f^3-a*b*d^4*e^
2*f-b^2*c^2*d^2*e*f^2+b^2*c*d^3*e^2*f)+b*d*(a^2*c^2*d^2*f^4-2*a^2*c*d^3*e*f^3+a^2*d^4*e^2*f^2-2*a*b*c^3*d*f^4+
4*a*b*c^2*d^2*e*f^3-2*a*b*c*d^3*e^2*f^2+b^2*c^4*f^4-2*b^2*c^3*d*e*f^3+b^2*c^2*d^2*e^2*f^2))*(f*x+e)^(3/2)+(a*d
*f-b*c*f)*(a^2*c^2*d^2*f^4-2*a^2*c*d^3*e*f^3+a^2*d^4*e^2*f^2-2*a*b*c^3*d*f^4+4*a*b*c^2*d^2*e*f^3-2*a*b*c*d^3*e
^2*f^2+b^2*c^4*f^4-2*b^2*c^3*d*e*f^3+b^2*c^2*d^2*e^2*f^2)*(f*x+e)^(1/2))-f^3*(a^3*c^3*d^3*f^3-3*a^3*c^2*d^4*e*
f^2+3*a^3*c*d^5*e^2*f-a^3*d^6*e^3-3*a^2*b*c^4*d^2*f^3+9*a^2*b*c^3*d^3*e*f^2-9*a^2*b*c^2*d^4*e^2*f+3*a^2*b*c*d^
5*e^3+3*a*b^2*c^5*d*f^3-9*a*b^2*c^4*d^2*e*f^2+9*a*b^2*c^3*d^3*e^2*f-3*a*b^2*c^2*d^4*e^3-b^3*c^6*f^3+3*b^3*c^5*
d*e*f^2-3*b^3*c^4*d^2*e^2*f+b^3*c^3*d^3*e^3)/d^6/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2
)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-%e*d>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (261) = 522\).
time = 0.86, size = 1742, normalized size = 6.22 \begin {gather*} \left [-\frac {3465 \, {\left ({\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} f^{5} - 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{4} e + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{3} e^{2}\right )} \sqrt {-\frac {c f - d e}{d}} \log \left (\frac {d f x - c f - 2 \, \sqrt {f x + e} d \sqrt {-\frac {c f - d e}{d}} + 2 \, d e}{d x + c}\right ) - 2 \, {\left (315 \, b^{3} d^{5} f^{5} x^{5} - 385 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{5} x^{4} + 495 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{5} x^{3} + 40 \, b^{3} d^{5} e^{5} - 693 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{5} x^{2} + 1155 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{5} x - 3465 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} f^{5} - 10 \, {\left (2 \, b^{3} d^{5} f x - 11 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f\right )} e^{4} + 5 \, {\left (3 \, b^{3} d^{5} f^{2} x^{2} - 11 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{2} x + 99 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{2}\right )} e^{3} + {\left (565 \, b^{3} d^{5} f^{3} x^{3} - 825 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{3} x^{2} + 1485 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{3} x - 5313 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{3}\right )} e^{2} + {\left (805 \, b^{3} d^{5} f^{4} x^{4} - 1045 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{4} x^{3} + 1485 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{4} x^{2} - 2541 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{4} x + 8085 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{4}\right )} e\right )} \sqrt {f x + e}}{3465 \, d^{6} f^{3}}, -\frac {2 \, {\left (3465 \, {\left ({\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} f^{5} - 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{4} e + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{3} e^{2}\right )} \sqrt {\frac {c f - d e}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {\frac {c f - d e}{d}}}{c f - d e}\right ) - {\left (315 \, b^{3} d^{5} f^{5} x^{5} - 385 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{5} x^{4} + 495 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{5} x^{3} + 40 \, b^{3} d^{5} e^{5} - 693 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{5} x^{2} + 1155 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{5} x - 3465 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} f^{5} - 10 \, {\left (2 \, b^{3} d^{5} f x - 11 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f\right )} e^{4} + 5 \, {\left (3 \, b^{3} d^{5} f^{2} x^{2} - 11 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{2} x + 99 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{2}\right )} e^{3} + {\left (565 \, b^{3} d^{5} f^{3} x^{3} - 825 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{3} x^{2} + 1485 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{3} x - 5313 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{3}\right )} e^{2} + {\left (805 \, b^{3} d^{5} f^{4} x^{4} - 1045 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} f^{4} x^{3} + 1485 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} f^{4} x^{2} - 2541 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} f^{4} x + 8085 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} f^{4}\right )} e\right )} \sqrt {f x + e}\right )}}{3465 \, d^{6} f^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="fricas")

[Out]

[-1/3465*(3465*((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5 - 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2
+ 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^4*e + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^3*e^2)*sqr
t(-(c*f - d*e)/d)*log((d*f*x - c*f - 2*sqrt(f*x + e)*d*sqrt(-(c*f - d*e)/d) + 2*d*e)/(d*x + c)) - 2*(315*b^3*d
^5*f^5*x^5 - 385*(b^3*c*d^4 - 3*a*b^2*d^5)*f^5*x^4 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5*x^3 +
40*b^3*d^5*e^5 - 693*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^5*x^2 + 1155*(b^3*c^4*d - 3*
a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5*x - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2
*d^3)*f^5 - 10*(2*b^3*d^5*f*x - 11*(b^3*c*d^4 - 3*a*b^2*d^5)*f)*e^4 + 5*(3*b^3*d^5*f^2*x^2 - 11*(b^3*c*d^4 - 3
*a*b^2*d^5)*f^2*x + 99*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^2)*e^3 + (565*b^3*d^5*f^3*x^3 - 825*(b^3*
c*d^4 - 3*a*b^2*d^5)*f^3*x^2 + 1485*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^3*x - 5313*(b^3*c^3*d^2 - 3*
a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^3)*e^2 + (805*b^3*d^5*f^4*x^4 - 1045*(b^3*c*d^4 - 3*a*b^2*d^5)*f^4*
x^3 + 1485*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^4*x^2 - 2541*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b
*c*d^4 - a^3*d^5)*f^4*x + 8085*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^4)*e)*sqrt(f*x +
e))/(d^6*f^3), -2/3465*(3465*((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5 - 2*(b^3*c^4*d - 3
*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^4*e + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5
)*f^3*e^2)*sqrt((c*f - d*e)/d)*arctan(-sqrt(f*x + e)*d*sqrt((c*f - d*e)/d)/(c*f - d*e)) - (315*b^3*d^5*f^5*x^5
- 385*(b^3*c*d^4 - 3*a*b^2*d^5)*f^5*x^4 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5*x^3 + 40*b^3*d^
5*e^5 - 693*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^5*x^2 + 1155*(b^3*c^4*d - 3*a*b^2*c^3*
d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5*x - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5
- 10*(2*b^3*d^5*f*x - 11*(b^3*c*d^4 - 3*a*b^2*d^5)*f)*e^4 + 5*(3*b^3*d^5*f^2*x^2 - 11*(b^3*c*d^4 - 3*a*b^2*d^5
)*f^2*x + 99*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^2)*e^3 + (565*b^3*d^5*f^3*x^3 - 825*(b^3*c*d^4 - 3*
a*b^2*d^5)*f^3*x^2 + 1485*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^3*x - 5313*(b^3*c^3*d^2 - 3*a*b^2*c^2*
d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^3)*e^2 + (805*b^3*d^5*f^4*x^4 - 1045*(b^3*c*d^4 - 3*a*b^2*d^5)*f^4*x^3 + 1485
*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^4*x^2 - 2541*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a
^3*d^5)*f^4*x + 8085*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^4)*e)*sqrt(f*x + e))/(d^6*f
^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (267) = 534\).
time = 112.74, size = 575, normalized size = 2.05 \begin {gather*} \frac {2 b^{3} \left (e + f x\right )^{\frac {11}{2}}}{11 d f^{3}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \cdot \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{9 d^{2} f^{3}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{7 d^{3} f^{3}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{5 d^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{3 d^{5}} + \frac {\sqrt {e + f x} \left (2 a^{3} c^{2} d^{3} f^{2} - 4 a^{3} c d^{4} e f + 2 a^{3} d^{5} e^{2} - 6 a^{2} b c^{3} d^{2} f^{2} + 12 a^{2} b c^{2} d^{3} e f - 6 a^{2} b c d^{4} e^{2} + 6 a b^{2} c^{4} d f^{2} - 12 a b^{2} c^{3} d^{2} e f + 6 a b^{2} c^{2} d^{3} e^{2} - 2 b^{3} c^{5} f^{2} + 4 b^{3} c^{4} d e f - 2 b^{3} c^{3} d^{2} e^{2}\right )}{d^{6}} - \frac {2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{7} \sqrt {\frac {c f - d e}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(f*x+e)**(5/2)/(d*x+c),x)

[Out]

2*b**3*(e + f*x)**(11/2)/(11*d*f**3) + (e + f*x)**(9/2)*(6*a*b**2*d*f - 2*b**3*c*f - 4*b**3*d*e)/(9*d**2*f**3)
+ (e + f*x)**(7/2)*(6*a**2*b*d**2*f**2 - 6*a*b**2*c*d*f**2 - 6*a*b**2*d**2*e*f + 2*b**3*c**2*f**2 + 2*b**3*c*
d*e*f + 2*b**3*d**2*e**2)/(7*d**3*f**3) + (e + f*x)**(5/2)*(2*a**3*d**3 - 6*a**2*b*c*d**2 + 6*a*b**2*c**2*d -
2*b**3*c**3)/(5*d**4) + (e + f*x)**(3/2)*(-2*a**3*c*d**3*f + 2*a**3*d**4*e + 6*a**2*b*c**2*d**2*f - 6*a**2*b*c
*d**3*e - 6*a*b**2*c**3*d*f + 6*a*b**2*c**2*d**2*e + 2*b**3*c**4*f - 2*b**3*c**3*d*e)/(3*d**5) + sqrt(e + f*x)
*(2*a**3*c**2*d**3*f**2 - 4*a**3*c*d**4*e*f + 2*a**3*d**5*e**2 - 6*a**2*b*c**3*d**2*f**2 + 12*a**2*b*c**2*d**3
*e*f - 6*a**2*b*c*d**4*e**2 + 6*a*b**2*c**4*d*f**2 - 12*a*b**2*c**3*d**2*e*f + 6*a*b**2*c**2*d**3*e**2 - 2*b**
3*c**5*f**2 + 4*b**3*c**4*d*e*f - 2*b**3*c**3*d**2*e**2)/d**6 - 2*(a*d - b*c)**3*(c*f - d*e)**3*atan(sqrt(e +
f*x)/sqrt((c*f - d*e)/d))/(d**7*sqrt((c*f - d*e)/d))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (261) = 522\).
time = 0.66, size = 1028, normalized size = 3.67 \begin {gather*} \frac {2 \, {\left (b^{3} c^{6} f^{3} - 3 \, a b^{2} c^{5} d f^{3} + 3 \, a^{2} b c^{4} d^{2} f^{3} - a^{3} c^{3} d^{3} f^{3} - 3 \, b^{3} c^{5} d f^{2} e + 9 \, a b^{2} c^{4} d^{2} f^{2} e - 9 \, a^{2} b c^{3} d^{3} f^{2} e + 3 \, a^{3} c^{2} d^{4} f^{2} e + 3 \, b^{3} c^{4} d^{2} f e^{2} - 9 \, a b^{2} c^{3} d^{3} f e^{2} + 9 \, a^{2} b c^{2} d^{4} f e^{2} - 3 \, a^{3} c d^{5} f e^{2} - b^{3} c^{3} d^{3} e^{3} + 3 \, a b^{2} c^{2} d^{4} e^{3} - 3 \, a^{2} b c d^{5} e^{3} + a^{3} d^{6} e^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{6}} + \frac {2 \, {\left (315 \, {\left (f x + e\right )}^{\frac {11}{2}} b^{3} d^{10} f^{30} - 385 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} c d^{9} f^{31} + 1155 \, {\left (f x + e\right )}^{\frac {9}{2}} a b^{2} d^{10} f^{31} + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c^{2} d^{8} f^{32} - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} c d^{9} f^{32} + 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a^{2} b d^{10} f^{32} - 693 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c^{3} d^{7} f^{33} + 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} c^{2} d^{8} f^{33} - 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} b c d^{9} f^{33} + 693 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{3} d^{10} f^{33} + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{4} d^{6} f^{34} - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{3} d^{7} f^{34} + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c^{2} d^{8} f^{34} - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} c d^{9} f^{34} - 3465 \, \sqrt {f x + e} b^{3} c^{5} d^{5} f^{35} + 10395 \, \sqrt {f x + e} a b^{2} c^{4} d^{6} f^{35} - 10395 \, \sqrt {f x + e} a^{2} b c^{3} d^{7} f^{35} + 3465 \, \sqrt {f x + e} a^{3} c^{2} d^{8} f^{35} - 770 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d^{10} f^{30} e + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c d^{9} f^{31} e - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} d^{10} f^{31} e - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{3} d^{7} f^{33} e + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{8} f^{33} e - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c d^{9} f^{33} e + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} d^{10} f^{33} e + 6930 \, \sqrt {f x + e} b^{3} c^{4} d^{6} f^{34} e - 20790 \, \sqrt {f x + e} a b^{2} c^{3} d^{7} f^{34} e + 20790 \, \sqrt {f x + e} a^{2} b c^{2} d^{8} f^{34} e - 6930 \, \sqrt {f x + e} a^{3} c d^{9} f^{34} e + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{10} f^{30} e^{2} - 3465 \, \sqrt {f x + e} b^{3} c^{3} d^{7} f^{33} e^{2} + 10395 \, \sqrt {f x + e} a b^{2} c^{2} d^{8} f^{33} e^{2} - 10395 \, \sqrt {f x + e} a^{2} b c d^{9} f^{33} e^{2} + 3465 \, \sqrt {f x + e} a^{3} d^{10} f^{33} e^{2}\right )}}{3465 \, d^{11} f^{33}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="giac")

[Out]

2*(b^3*c^6*f^3 - 3*a*b^2*c^5*d*f^3 + 3*a^2*b*c^4*d^2*f^3 - a^3*c^3*d^3*f^3 - 3*b^3*c^5*d*f^2*e + 9*a*b^2*c^4*d
^2*f^2*e - 9*a^2*b*c^3*d^3*f^2*e + 3*a^3*c^2*d^4*f^2*e + 3*b^3*c^4*d^2*f*e^2 - 9*a*b^2*c^3*d^3*f*e^2 + 9*a^2*b
*c^2*d^4*f*e^2 - 3*a^3*c*d^5*f*e^2 - b^3*c^3*d^3*e^3 + 3*a*b^2*c^2*d^4*e^3 - 3*a^2*b*c*d^5*e^3 + a^3*d^6*e^3)*
arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^6) + 2/3465*(315*(f*x + e)^(11/2)*b^3*d^10*
f^30 - 385*(f*x + e)^(9/2)*b^3*c*d^9*f^31 + 1155*(f*x + e)^(9/2)*a*b^2*d^10*f^31 + 495*(f*x + e)^(7/2)*b^3*c^2
*d^8*f^32 - 1485*(f*x + e)^(7/2)*a*b^2*c*d^9*f^32 + 1485*(f*x + e)^(7/2)*a^2*b*d^10*f^32 - 693*(f*x + e)^(5/2)
*b^3*c^3*d^7*f^33 + 2079*(f*x + e)^(5/2)*a*b^2*c^2*d^8*f^33 - 2079*(f*x + e)^(5/2)*a^2*b*c*d^9*f^33 + 693*(f*x
+ e)^(5/2)*a^3*d^10*f^33 + 1155*(f*x + e)^(3/2)*b^3*c^4*d^6*f^34 - 3465*(f*x + e)^(3/2)*a*b^2*c^3*d^7*f^34 +
3465*(f*x + e)^(3/2)*a^2*b*c^2*d^8*f^34 - 1155*(f*x + e)^(3/2)*a^3*c*d^9*f^34 - 3465*sqrt(f*x + e)*b^3*c^5*d^5
*f^35 + 10395*sqrt(f*x + e)*a*b^2*c^4*d^6*f^35 - 10395*sqrt(f*x + e)*a^2*b*c^3*d^7*f^35 + 3465*sqrt(f*x + e)*a
^3*c^2*d^8*f^35 - 770*(f*x + e)^(9/2)*b^3*d^10*f^30*e + 495*(f*x + e)^(7/2)*b^3*c*d^9*f^31*e - 1485*(f*x + e)^
(7/2)*a*b^2*d^10*f^31*e - 1155*(f*x + e)^(3/2)*b^3*c^3*d^7*f^33*e + 3465*(f*x + e)^(3/2)*a*b^2*c^2*d^8*f^33*e
- 3465*(f*x + e)^(3/2)*a^2*b*c*d^9*f^33*e + 1155*(f*x + e)^(3/2)*a^3*d^10*f^33*e + 6930*sqrt(f*x + e)*b^3*c^4*
d^6*f^34*e - 20790*sqrt(f*x + e)*a*b^2*c^3*d^7*f^34*e + 20790*sqrt(f*x + e)*a^2*b*c^2*d^8*f^34*e - 6930*sqrt(f
*x + e)*a^3*c*d^9*f^34*e + 495*(f*x + e)^(7/2)*b^3*d^10*f^30*e^2 - 3465*sqrt(f*x + e)*b^3*c^3*d^7*f^33*e^2 + 1
0395*sqrt(f*x + e)*a*b^2*c^2*d^8*f^33*e^2 - 10395*sqrt(f*x + e)*a^2*b*c*d^9*f^33*e^2 + 3465*sqrt(f*x + e)*a^3*
d^10*f^33*e^2)/(d^11*f^33)

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Mupad [B]
time = 0.16, size = 897, normalized size = 3.20 \begin {gather*} {\left (e+f\,x\right )}^{7/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{7\,d\,f^3}\right )-{\left (e+f\,x\right )}^{9/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{9\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{9\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{5\,d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{11/2}}{11\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{-a^3\,c^3\,d^3\,f^3+3\,a^3\,c^2\,d^4\,e\,f^2-3\,a^3\,c\,d^5\,e^2\,f+a^3\,d^6\,e^3+3\,a^2\,b\,c^4\,d^2\,f^3-9\,a^2\,b\,c^3\,d^3\,e\,f^2+9\,a^2\,b\,c^2\,d^4\,e^2\,f-3\,a^2\,b\,c\,d^5\,e^3-3\,a\,b^2\,c^5\,d\,f^3+9\,a\,b^2\,c^4\,d^2\,e\,f^2-9\,a\,b^2\,c^3\,d^3\,e^2\,f+3\,a\,b^2\,c^2\,d^4\,e^3+b^3\,c^6\,f^3-3\,b^3\,c^5\,d\,e\,f^2+3\,b^3\,c^4\,d^2\,e^2\,f-b^3\,c^3\,d^3\,e^3}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{d^{13/2}}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,{\left (c\,f^4-d\,e\,f^3\right )}^2}{d^2\,f^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^(5/2)*(a + b*x)^3)/(c + d*x),x)

[Out]

(e + f*x)^(7/2)*((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(7*
d*f^3) + (6*b*(a*f - b*e)^2)/(7*d*f^3)) - (e + f*x)^(9/2)*((6*b^3*e - 6*a*b^2*f)/(9*d*f^3) + (2*b^3*(c*f^4 - d
*e*f^3))/(9*d^2*f^6)) + (e + f*x)^(5/2)*((2*(a*f - b*e)^3)/(5*d*f^3) - (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*
b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(d*f^3))*(c*f^4 - d*e*f^3))
/(5*d*f^3)) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^3*(c*f - d*e)
^(5/2))/(a^3*d^6*e^3 + b^3*c^6*f^3 - a^3*c^3*d^3*f^3 - b^3*c^3*d^3*e^3 - 3*a^2*b*c*d^5*e^3 - 3*a*b^2*c^5*d*f^3
- 3*a^3*c*d^5*e^2*f - 3*b^3*c^5*d*e*f^2 + 3*a*b^2*c^2*d^4*e^3 + 3*a^2*b*c^4*d^2*f^3 + 3*a^3*c^2*d^4*e*f^2 + 3
*b^3*c^4*d^2*e^2*f - 9*a*b^2*c^3*d^3*e^2*f + 9*a*b^2*c^4*d^2*e*f^2 + 9*a^2*b*c^2*d^4*e^2*f - 9*a^2*b*c^3*d^3*e
*f^2))*(a*d - b*c)^3*(c*f - d*e)^(5/2))/d^(13/2) - ((e + f*x)^(3/2)*((2*(a*f - b*e)^3)/(d*f^3) - (((((6*b^3*e
- 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(
d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3))/(3*d*f^3) + ((e + f*x)^(1/2)*((2*(a*f - b*e)^3)/(d*f^3)
- (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*
(a*f - b*e)^2)/(d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3)^2)/(d^2*f^6)

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