3.18.74 \(\int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx\) [1774]

Optimal. Leaf size=208 \[ \frac {2 (b c-a d)^2 (d e-c f)^2 \sqrt {e+f x}}{d^5}+\frac {2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac {2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac {2 b^2 (e+f x)^{9/2}}{9 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 208, 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 c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}+\frac {2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac {2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}-\frac {2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac {2 b^2 (e+f x)^{9/2}}{9 d f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.44, size = 295, normalized size = 1.42 \begin {gather*} \frac {2 \sqrt {e+f x} \left (21 a^2 d^2 f^2 \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 )+6 a b d f \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 )+b^2 \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 )\right )}{315 d^5 f^2}-\frac {2 (b c-a d)^2 (-d e+c f)^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e + f*x]*(21*a^2*d^2*f^2*(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)) + 6*
a*b*d*f*(-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)) + b^2*(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))))/(315*d^5*f^2) - (2*(b*c - a*d)^2*(-(d*e) + c*f)^(5/2)*Arc
Tan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(11/2)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(626\) vs. \(2(180)=360\).
time = 0.09, size = 627, normalized size = 3.01 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^2*(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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (191) = 382\).
time = 1.11, size = 1071, normalized size = 5.15 \begin {gather*} \left [\frac {315 \, {\left ({\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f^{4} - 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{3} e + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{2} 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 (35 \, b^{2} d^{4} f^{4} x^{4} - 45 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{4} x^{3} + 63 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{4} x^{2} - 10 \, b^{2} d^{4} e^{4} - 105 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{4} x + 315 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f^{4} + 5 \, {\left (b^{2} d^{4} f x - 9 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f\right )} e^{3} + 3 \, {\left (25 \, b^{2} d^{4} f^{2} x^{2} - 45 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{2} x + 161 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{2}\right )} e^{2} + {\left (95 \, b^{2} d^{4} f^{3} x^{3} - 135 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{3} x^{2} + 231 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{3} x - 735 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}}{315 \, d^{5} f^{2}}, \frac {2 \, {\left (315 \, {\left ({\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f^{4} - 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{3} e + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{2} 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 (35 \, b^{2} d^{4} f^{4} x^{4} - 45 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{4} x^{3} + 63 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{4} x^{2} - 10 \, b^{2} d^{4} e^{4} - 105 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{4} x + 315 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f^{4} + 5 \, {\left (b^{2} d^{4} f x - 9 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f\right )} e^{3} + 3 \, {\left (25 \, b^{2} d^{4} f^{2} x^{2} - 45 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{2} x + 161 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{2}\right )} e^{2} + {\left (95 \, b^{2} d^{4} f^{3} x^{3} - 135 \, {\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} f^{3} x^{2} + 231 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} f^{3} x - 735 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}\right )}}{315 \, d^{5} f^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/315*(315*((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 - 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^3*e + (b^
2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*e^2)*sqrt(-(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*(35*b^2*d^4*f^4*x^4 - 45*(b^2*c*d^3 - 2*a*b*d^4)*f^4*x^3 + 63*(b^2*c^2*d^
2 - 2*a*b*c*d^3 + a^2*d^4)*f^4*x^2 - 10*b^2*d^4*e^4 - 105*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^4*x + 315*
(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 + 5*(b^2*d^4*f*x - 9*(b^2*c*d^3 - 2*a*b*d^4)*f)*e^3 + 3*(25*b^2*d^4*
f^2*x^2 - 45*(b^2*c*d^3 - 2*a*b*d^4)*f^2*x + 161*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2)*e^2 + (95*b^2*d^4*
f^3*x^3 - 135*(b^2*c*d^3 - 2*a*b*d^4)*f^3*x^2 + 231*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^3*x - 735*(b^2*c^3
*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^3)*e)*sqrt(f*x + e))/(d^5*f^2), 2/315*(315*((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2
*d^2)*f^4 - 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^3*e + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*e^2)*s
qrt((c*f - d*e)/d)*arctan(-sqrt(f*x + e)*d*sqrt((c*f - d*e)/d)/(c*f - d*e)) + (35*b^2*d^4*f^4*x^4 - 45*(b^2*c*
d^3 - 2*a*b*d^4)*f^4*x^3 + 63*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^4*x^2 - 10*b^2*d^4*e^4 - 105*(b^2*c^3*d
- 2*a*b*c^2*d^2 + a^2*c*d^3)*f^4*x + 315*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 + 5*(b^2*d^4*f*x - 9*(b^2*c
*d^3 - 2*a*b*d^4)*f)*e^3 + 3*(25*b^2*d^4*f^2*x^2 - 45*(b^2*c*d^3 - 2*a*b*d^4)*f^2*x + 161*(b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*f^2)*e^2 + (95*b^2*d^4*f^3*x^3 - 135*(b^2*c*d^3 - 2*a*b*d^4)*f^3*x^2 + 231*(b^2*c^2*d^2 - 2*
a*b*c*d^3 + a^2*d^4)*f^3*x - 735*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^3)*e)*sqrt(f*x + e))/(d^5*f^2)]

________________________________________________________________________________________

Sympy [A]
time = 93.61, size = 374, normalized size = 1.80 \begin {gather*} \frac {2 b^{2} \left (e + f x\right )^{\frac {9}{2}}}{9 d f^{2}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{7 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{5 d^{3}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{3 d^{4}} + \frac {\sqrt {e + f x} \left (2 a^{2} c^{2} d^{2} f^{2} - 4 a^{2} c d^{3} e f + 2 a^{2} d^{4} e^{2} - 4 a b c^{3} d f^{2} + 8 a b c^{2} d^{2} e f - 4 a b c d^{3} e^{2} + 2 b^{2} c^{4} f^{2} - 4 b^{2} c^{3} d e f + 2 b^{2} c^{2} d^{2} e^{2}\right )}{d^{5}} - \frac {2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{6} \sqrt {\frac {c f - d e}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*(e + f*x)**(9/2)/(9*d*f**2) + (e + f*x)**(7/2)*(4*a*b*d*f - 2*b**2*c*f - 2*b**2*d*e)/(7*d**2*f**2) + (e
+ f*x)**(5/2)*(2*a**2*d**2 - 4*a*b*c*d + 2*b**2*c**2)/(5*d**3) + (e + f*x)**(3/2)*(-2*a**2*c*d**2*f + 2*a**2*
d**3*e + 4*a*b*c**2*d*f - 4*a*b*c*d**2*e - 2*b**2*c**3*f + 2*b**2*c**2*d*e)/(3*d**4) + sqrt(e + f*x)*(2*a**2*c
**2*d**2*f**2 - 4*a**2*c*d**3*e*f + 2*a**2*d**4*e**2 - 4*a*b*c**3*d*f**2 + 8*a*b*c**2*d**2*e*f - 4*a*b*c*d**3*
e**2 + 2*b**2*c**4*f**2 - 4*b**2*c**3*d*e*f + 2*b**2*c**2*d**2*e**2)/d**5 - 2*(a*d - b*c)**2*(c*f - d*e)**3*at
an(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**6*sqrt((c*f - d*e)/d))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (191) = 382\).
time = 0.63, size = 673, normalized size = 3.24 \begin {gather*} -\frac {2 \, {\left (b^{2} c^{5} f^{3} - 2 \, a b c^{4} d f^{3} + a^{2} c^{3} d^{2} f^{3} - 3 \, b^{2} c^{4} d f^{2} e + 6 \, a b c^{3} d^{2} f^{2} e - 3 \, a^{2} c^{2} d^{3} f^{2} e + 3 \, b^{2} c^{3} d^{2} f e^{2} - 6 \, a b c^{2} d^{3} f e^{2} + 3 \, a^{2} c d^{4} f e^{2} - b^{2} c^{2} d^{3} e^{3} + 2 \, a b c d^{4} e^{3} - a^{2} d^{5} 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^{5}} + \frac {2 \, {\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{2} d^{8} f^{16} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} c d^{7} f^{17} + 90 \, {\left (f x + e\right )}^{\frac {7}{2}} a b d^{8} f^{17} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} c^{2} d^{6} f^{18} - 126 \, {\left (f x + e\right )}^{\frac {5}{2}} a b c d^{7} f^{18} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} d^{8} f^{18} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{3} d^{5} f^{19} + 210 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c^{2} d^{6} f^{19} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} c d^{7} f^{19} + 315 \, \sqrt {f x + e} b^{2} c^{4} d^{4} f^{20} - 630 \, \sqrt {f x + e} a b c^{3} d^{5} f^{20} + 315 \, \sqrt {f x + e} a^{2} c^{2} d^{6} f^{20} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} d^{8} f^{16} e + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} d^{6} f^{18} e - 210 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c d^{7} f^{18} e + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{8} f^{18} e - 630 \, \sqrt {f x + e} b^{2} c^{3} d^{5} f^{19} e + 1260 \, \sqrt {f x + e} a b c^{2} d^{6} f^{19} e - 630 \, \sqrt {f x + e} a^{2} c d^{7} f^{19} e + 315 \, \sqrt {f x + e} b^{2} c^{2} d^{6} f^{18} e^{2} - 630 \, \sqrt {f x + e} a b c d^{7} f^{18} e^{2} + 315 \, \sqrt {f x + e} a^{2} d^{8} f^{18} e^{2}\right )}}{315 \, d^{9} f^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b^2*c^5*f^3 - 2*a*b*c^4*d*f^3 + a^2*c^3*d^2*f^3 - 3*b^2*c^4*d*f^2*e + 6*a*b*c^3*d^2*f^2*e - 3*a^2*c^2*d^3*
f^2*e + 3*b^2*c^3*d^2*f*e^2 - 6*a*b*c^2*d^3*f*e^2 + 3*a^2*c*d^4*f*e^2 - b^2*c^2*d^3*e^3 + 2*a*b*c*d^4*e^3 - a^
2*d^5*e^3)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^5) + 2/315*(35*(f*x + e)^(9/2)*b
^2*d^8*f^16 - 45*(f*x + e)^(7/2)*b^2*c*d^7*f^17 + 90*(f*x + e)^(7/2)*a*b*d^8*f^17 + 63*(f*x + e)^(5/2)*b^2*c^2
*d^6*f^18 - 126*(f*x + e)^(5/2)*a*b*c*d^7*f^18 + 63*(f*x + e)^(5/2)*a^2*d^8*f^18 - 105*(f*x + e)^(3/2)*b^2*c^3
*d^5*f^19 + 210*(f*x + e)^(3/2)*a*b*c^2*d^6*f^19 - 105*(f*x + e)^(3/2)*a^2*c*d^7*f^19 + 315*sqrt(f*x + e)*b^2*
c^4*d^4*f^20 - 630*sqrt(f*x + e)*a*b*c^3*d^5*f^20 + 315*sqrt(f*x + e)*a^2*c^2*d^6*f^20 - 45*(f*x + e)^(7/2)*b^
2*d^8*f^16*e + 105*(f*x + e)^(3/2)*b^2*c^2*d^6*f^18*e - 210*(f*x + e)^(3/2)*a*b*c*d^7*f^18*e + 105*(f*x + e)^(
3/2)*a^2*d^8*f^18*e - 630*sqrt(f*x + e)*b^2*c^3*d^5*f^19*e + 1260*sqrt(f*x + e)*a*b*c^2*d^6*f^19*e - 630*sqrt(
f*x + e)*a^2*c*d^7*f^19*e + 315*sqrt(f*x + e)*b^2*c^2*d^6*f^18*e^2 - 630*sqrt(f*x + e)*a*b*c*d^7*f^18*e^2 + 31
5*sqrt(f*x + e)*a^2*d^8*f^18*e^2)/(d^9*f^18)

________________________________________________________________________________________

Mupad [B]
time = 0.16, size = 602, normalized size = 2.89 \begin {gather*} {\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{5\,d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d\,f^2}\right )-{\left (e+f\,x\right )}^{7/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{7\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{7\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{9/2}}{9\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{5/2}}{-a^2\,c^3\,d^2\,f^3+3\,a^2\,c^2\,d^3\,e\,f^2-3\,a^2\,c\,d^4\,e^2\,f+a^2\,d^5\,e^3+2\,a\,b\,c^4\,d\,f^3-6\,a\,b\,c^3\,d^2\,e\,f^2+6\,a\,b\,c^2\,d^3\,e^2\,f-2\,a\,b\,c\,d^4\,e^3-b^2\,c^5\,f^3+3\,b^2\,c^4\,d\,e\,f^2-3\,b^2\,c^3\,d^2\,e^2\,f+b^2\,c^2\,d^3\,e^3}\right )\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{5/2}}{d^{11/2}}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d\,f^2}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,{\left (c\,f^3-d\,e\,f^2\right )}^2}{d^2\,f^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________