3.18.84 \(\int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx\) [1784]

Optimal. Leaf size=210 \[ -\frac {2 (b c-a d)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, 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)^{3/2} \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 )}{3 d^3 f^3}-\frac {2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3}{d^4}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.37, size = 249, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3+105 a^2 b d^2 f^2 (-3 c f+d (e+f x))-21 a b^2 d f \left (-15 c^2 f^2+5 c d f (e+f x)+d^2 \left (2 e^2-e f x-3 f^2 x^2\right )\right )+b^3 \left (-105 c^3 f^3+35 c^2 d f^2 (e+f x)-7 c d^2 f \left (-2 e^2+e f x+3 f^2 x^2\right )+d^3 \left (8 e^3-4 e^2 f x+3 e f^2 x^2+15 f^3 x^3\right )\right )\right )}{105 d^4 f^3}-\frac {2 (-b c+a d)^3 \sqrt {-d e+c f} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3 + 105*a^2*b*d^2*f^2*(-3*c*f + d*(e + f*x)) - 21*a*b^2*d*f*(-15*c^2*f^2 + 5*c
*d*f*(e + f*x) + d^2*(2*e^2 - e*f*x - 3*f^2*x^2)) + b^3*(-105*c^3*f^3 + 35*c^2*d*f^2*(e + f*x) - 7*c*d^2*f*(-2
*e^2 + e*f*x + 3*f^2*x^2) + d^3*(8*e^3 - 4*e^2*f*x + 3*e*f^2*x^2 + 15*f^3*x^3))))/(105*d^4*f^3) - (2*(-(b*c) +
a*d)^3*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(9/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs. \(2(186)=372\).
time = 0.10, size = 390, normalized size = 1.86

method	result	size
derivativedivides	\(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} d^{4} e -3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\)	\(390\)
default	\(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} d^{4} e -3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\)	\(390\)
risch	\(\frac {2 \left (15 b^{3} d^{3} f^{3} x^{3}+63 a \,b^{2} d^{3} f^{3} x^{2}-21 b^{3} c \,d^{2} f^{3} x^{2}+3 b^{3} d^{3} e \,f^{2} x^{2}+105 a^{2} b \,d^{3} f^{3} x -105 a \,b^{2} c \,d^{2} f^{3} x +21 a \,b^{2} d^{3} e \,f^{2} x +35 b^{3} c^{2} d \,f^{3} x -7 b^{3} c \,d^{2} e \,f^{2} x -4 b^{3} d^{3} e^{2} f x +105 a^{3} d^{3} f^{3}-315 a^{2} b c \,d^{2} f^{3}+105 a^{2} b \,d^{3} e \,f^{2}+315 a \,b^{2} c^{2} d \,f^{3}-105 a \,b^{2} c \,d^{2} e \,f^{2}-42 a \,b^{2} d^{3} e^{2} f -105 b^{3} c^{3} f^{3}+35 b^{3} c^{2} d e \,f^{2}+14 b^{3} c \,d^{2} e^{2} f +8 b^{3} d^{3} e^{3}\right ) \sqrt {f x +e}}{105 f^{3} d^{4}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3} c f}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3} e}{\sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b \,c^{2} f}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b c e}{d \sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{3} f}{d^{3} \sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{2} e}{d^{2} \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{4} f}{d^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{3} e}{d^{3} \sqrt {\left (c f -d e \right ) d}}\)	\(645\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(f*x+e)^(1/2)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

2/f^3*(1/d^4*(1/7*b^3*(f*x+e)^(7/2)*d^3+3/5*a*b^2*d^3*f*(f*x+e)^(5/2)-1/5*b^3*c*d^2*f*(f*x+e)^(5/2)-2/5*b^3*d^
3*e*(f*x+e)^(5/2)+a^2*b*d^3*f^2*(f*x+e)^(3/2)-a*b^2*c*d^2*f^2*(f*x+e)^(3/2)-a*b^2*d^3*e*f*(f*x+e)^(3/2)+1/3*b^
3*c^2*d*f^2*(f*x+e)^(3/2)+1/3*b^3*c*d^2*e*f*(f*x+e)^(3/2)+1/3*b^3*d^3*e^2*(f*x+e)^(3/2)+a^3*d^3*f^3*(f*x+e)^(1
/2)-3*a^2*b*c*d^2*f^3*(f*x+e)^(1/2)+3*a*b^2*c^2*d*f^3*(f*x+e)^(1/2)-b^3*c^3*f^3*(f*x+e)^(1/2))-f^3*(a^3*c*d^3*
f-a^3*d^4*e-3*a^2*b*c^2*d^2*f+3*a^2*b*c*d^3*e+3*a*b^2*c^3*d*f-3*a*b^2*c^2*d^2*e-b^3*c^4*f+b^3*c^3*d*e)/d^4/((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)^(1/2)/(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 [A]
time = 1.44, size = 714, normalized size = 3.40 \begin {gather*} \left [-\frac {105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \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 (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} - 21 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3} x^{2} + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3} x - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} - 2 \, {\left (2 \, b^{3} d^{3} f x - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} e^{2} + {\left (3 \, b^{3} d^{3} f^{2} x^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{2} x + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}}{105 \, d^{4} f^{3}}, -\frac {2 \, {\left (105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \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 (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} - 21 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3} x^{2} + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3} x - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} - 2 \, {\left (2 \, b^{3} d^{3} f x - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} e^{2} + {\left (3 \, b^{3} d^{3} f^{2} x^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{2} x + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(1/2)/(d*x+c),x, algorithm="fricas")

[Out]

[-1/105*(105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*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*(15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3 - 21*(b^3*c*
d^2 - 3*a*b^2*d^3)*f^3*x^2 + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^3*x - 105*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*f^3 - 2*(2*b^3*d^3*f*x - 7*(b^3*c*d^2 - 3*a*b^2*d^3)*f)*e^2 + (3*b^3*d^3*f^2*x^2 -
7*(b^3*c*d^2 - 3*a*b^2*d^3)*f^2*x + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^2)*e)*sqrt(f*x + e))/(d^4*
f^3), -2/105*(105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*sqrt((c*f - d*e)/d)*arctan(-sqrt(f*x
+ e)*d*sqrt((c*f - d*e)/d)/(c*f - d*e)) - (15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3 - 21*(b^3*c*d^2 - 3*a*b^2*d^3)*
f^3*x^2 + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^3*x - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 -
a^3*d^3)*f^3 - 2*(2*b^3*d^3*f*x - 7*(b^3*c*d^2 - 3*a*b^2*d^3)*f)*e^2 + (3*b^3*d^3*f^2*x^2 - 7*(b^3*c*d^2 - 3*a
*b^2*d^3)*f^2*x + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^2)*e)*sqrt(f*x + e))/(d^4*f^3)]

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Sympy [A]
time = 9.05, size = 269, normalized size = 1.28 \begin {gather*} \frac {2 \left (\frac {b^{3} \left (e + f x\right )^{\frac {7}{2}}}{7 d f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{5 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{3 d^{3} f^{2}} + \frac {\sqrt {e + f x} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{d^{4}} - \frac {f \left (a d - b c\right )^{3} \left (c f - d e\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}}\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(f*x+e)**(1/2)/(d*x+c),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (195) = 390\).
time = 0.51, size = 436, normalized size = 2.08 \begin {gather*} \frac {2 \, {\left (b^{3} c^{4} f - 3 \, a b^{2} c^{3} d f + 3 \, a^{2} b c^{2} d^{2} f - a^{3} c d^{3} f - b^{3} c^{3} d e + 3 \, a b^{2} c^{2} d^{2} e - 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e\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^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{6} f^{18} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c d^{5} f^{19} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} d^{6} f^{19} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} f^{20} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c d^{5} f^{20} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d^{6} f^{20} - 105 \, \sqrt {f x + e} b^{3} c^{3} d^{3} f^{21} + 315 \, \sqrt {f x + e} a b^{2} c^{2} d^{4} f^{21} - 315 \, \sqrt {f x + e} a^{2} b c d^{5} f^{21} + 105 \, \sqrt {f x + e} a^{3} d^{6} f^{21} - 42 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{6} f^{18} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c d^{5} f^{19} e - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d^{6} f^{19} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{6} f^{18} e^{2}\right )}}{105 \, d^{7} f^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(1/2)/(d*x+c),x, algorithm="giac")

[Out]

2*(b^3*c^4*f - 3*a*b^2*c^3*d*f + 3*a^2*b*c^2*d^2*f - a^3*c*d^3*f - b^3*c^3*d*e + 3*a*b^2*c^2*d^2*e - 3*a^2*b*c
*d^3*e + a^3*d^4*e)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^4) + 2/105*(15*(f*x + e
)^(7/2)*b^3*d^6*f^18 - 21*(f*x + e)^(5/2)*b^3*c*d^5*f^19 + 63*(f*x + e)^(5/2)*a*b^2*d^6*f^19 + 35*(f*x + e)^(3
/2)*b^3*c^2*d^4*f^20 - 105*(f*x + e)^(3/2)*a*b^2*c*d^5*f^20 + 105*(f*x + e)^(3/2)*a^2*b*d^6*f^20 - 105*sqrt(f*
x + e)*b^3*c^3*d^3*f^21 + 315*sqrt(f*x + e)*a*b^2*c^2*d^4*f^21 - 315*sqrt(f*x + e)*a^2*b*c*d^5*f^21 + 105*sqrt
(f*x + e)*a^3*d^6*f^21 - 42*(f*x + e)^(5/2)*b^3*d^6*f^18*e + 35*(f*x + e)^(3/2)*b^3*c*d^5*f^19*e - 105*(f*x +
e)^(3/2)*a*b^2*d^6*f^19*e + 35*(f*x + e)^(3/2)*b^3*d^6*f^18*e^2)/(d^7*f^21)

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Mupad [B]
time = 0.14, size = 451, normalized size = 2.15 \begin {gather*} {\left (e+f\,x\right )}^{3/2}\,\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 )}{3\,d\,f^3}+\frac {2\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{5\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d^2\,f^6}\right )+\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 )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,1{}\mathrm {i}}{-f\,a^3\,c\,d^3+e\,a^3\,d^4+3\,f\,a^2\,b\,c^2\,d^2-3\,e\,a^2\,b\,c\,d^3-3\,f\,a\,b^2\,c^3\,d+3\,e\,a\,b^2\,c^2\,d^2+f\,b^3\,c^4-e\,b^3\,c^3\,d}\right )\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,2{}\mathrm {i}}{d^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^(1/2)*(a + b*x)^3)/(c + d*x),x)

[Out]

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

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