Optimal. Leaf size=185 \[ -\frac {3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac {3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac {d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac {(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac {A d^3}{2 b^3 x^2}-\frac {(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)}-\frac {(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac {d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac {3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac {3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac {A d^3}{2 b^3 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {A d^3}{b^3 x^3}+\frac {d^2 (b B d-3 A c d+3 A b e)}{b^4 x^2}+\frac {3 d (-c d+b e) (b B d-2 A c d+A b e)}{b^5 x}-\frac {(b B-A c) (-c d+b e)^3}{b^3 c (b+c x)^3}+\frac {(-c d+b e)^2 \left (2 b B c d-3 A c^2 d+b^2 B e\right )}{b^4 c (b+c x)^2}-\frac {3 c d (-c d+b e) (b B d-2 A c d+A b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {A d^3}{2 b^3 x^2}-\frac {d^2 (b B d-3 A c d+3 A b e)}{b^4 x}-\frac {(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac {(c d-b e)^2 \left (2 b B c d-3 A c^2 d+b^2 B e\right )}{b^4 c^2 (b+c x)}-\frac {3 d (c d-b e) (b B d-2 A c d+A b e) \log (x)}{b^5}+\frac {3 d (c d-b e) (b B d-2 A c d+A b e) \log (b+c x)}{b^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 177, normalized size = 0.96 \begin {gather*} -\frac {\frac {2 b (c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{c^2 (b+c x)}-\frac {b^2 (b B-A c) (b e-c d)^3}{c^2 (b+c x)^2}+\frac {A b^2 d^3}{x^2}+\frac {2 b d^2 (3 A b e-3 A c d+b B d)}{x}-6 d \log (x) (b e-c d) (A b e-2 A c d+b B d)+6 d (b e-c d) \log (b+c x) (A b e-2 A c d+b B d)}{2 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 627, normalized size = 3.39 \begin {gather*} -\frac {A b^{4} c^{2} d^{3} - 2 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 3 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (9 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} - 9 \, {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e + 3 \, {\left (B b^{5} c - 3 \, A b^{4} c^{2}\right )} d e^{2} + {\left (B b^{6} + A b^{5} c\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{4} c^{2} d^{2} e + {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d^{3}\right )} x + 6 \, {\left ({\left (A b^{2} c^{4} d e^{2} - {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} + {\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \, {\left (A b^{3} c^{3} d e^{2} - {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (A b^{4} c^{2} d e^{2} - {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} + {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \, {\left ({\left (A b^{2} c^{4} d e^{2} - {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} + {\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \, {\left (A b^{3} c^{3} d e^{2} - {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (A b^{4} c^{2} d e^{2} - {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} + {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 391, normalized size = 2.11 \begin {gather*} -\frac {3 \, {\left (B b c d^{3} - 2 \, A c^{2} d^{3} - B b^{2} d^{2} e + 3 \, A b c d^{2} e - A b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {3 \, {\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{4} d^{3} x^{3} - 12 \, A c^{5} d^{3} x^{3} - 6 \, B b^{2} c^{3} d^{2} x^{3} e + 18 \, A b c^{4} d^{2} x^{3} e + 9 \, B b^{2} c^{3} d^{3} x^{2} - 18 \, A b c^{4} d^{3} x^{2} - 6 \, A b^{2} c^{3} d x^{3} e^{2} - 9 \, B b^{3} c^{2} d^{2} x^{2} e + 27 \, A b^{2} c^{3} d^{2} x^{2} e + 2 \, B b^{3} c^{2} d^{3} x - 4 \, A b^{2} c^{3} d^{3} x + 2 \, B b^{4} c x^{3} e^{3} + 3 \, B b^{4} c d x^{2} e^{2} - 9 \, A b^{3} c^{2} d x^{2} e^{2} + 6 \, A b^{3} c^{2} d^{2} x e + A b^{3} c^{2} d^{3} + B b^{5} x^{2} e^{3} + A b^{4} c x^{2} e^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 440, normalized size = 2.38 \begin {gather*} \frac {3 A d \,e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {3 A c \,d^{2} e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {A \,c^{2} d^{3}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {A \,e^{3}}{2 \left (c x +b \right )^{2} c}+\frac {B b \,e^{3}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {3 B \,d^{2} e}{2 \left (c x +b \right )^{2} b}-\frac {B c \,d^{3}}{2 \left (c x +b \right )^{2} b^{2}}-\frac {3 B d \,e^{2}}{2 \left (c x +b \right )^{2} c}+\frac {3 A d \,e^{2}}{\left (c x +b \right ) b^{2}}-\frac {6 A c \,d^{2} e}{\left (c x +b \right ) b^{3}}+\frac {3 A d \,e^{2} \ln \relax (x )}{b^{3}}-\frac {3 A d \,e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 A \,c^{2} d^{3}}{\left (c x +b \right ) b^{4}}-\frac {9 A c \,d^{2} e \ln \relax (x )}{b^{4}}+\frac {9 A c \,d^{2} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 A \,c^{2} d^{3} \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} d^{3} \ln \left (c x +b \right )}{b^{5}}+\frac {3 B \,d^{2} e}{\left (c x +b \right ) b^{2}}-\frac {2 B c \,d^{3}}{\left (c x +b \right ) b^{3}}+\frac {3 B \,d^{2} e \ln \relax (x )}{b^{3}}-\frac {3 B \,d^{2} e \ln \left (c x +b \right )}{b^{3}}-\frac {3 B c \,d^{3} \ln \relax (x )}{b^{4}}+\frac {3 B c \,d^{3} \ln \left (c x +b \right )}{b^{4}}-\frac {B \,e^{3}}{\left (c x +b \right ) c^{2}}-\frac {3 A \,d^{2} e}{b^{3} x}+\frac {3 A c \,d^{3}}{b^{4} x}-\frac {B \,d^{3}}{b^{3} x}-\frac {A \,d^{3}}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 347, normalized size = 1.88 \begin {gather*} -\frac {A b^{3} c^{2} d^{3} - 2 \, {\left (3 \, A b^{2} c^{3} d e^{2} - B b^{4} c e^{3} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + 3 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{2} e\right )} x^{3} + {\left (9 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 9 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{2} e + 3 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2} + {\left (B b^{5} + A b^{4} c\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{3} c^{2} d^{2} e + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x}{2 \, {\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac {3 \, {\left (A b^{2} d e^{2} - {\left (B b c - 2 \, A c^{2}\right )} d^{3} + {\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (A b^{2} d e^{2} - {\left (B b c - 2 \, A c^{2}\right )} d^{3} + {\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \relax (x)}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.62, size = 345, normalized size = 1.86 \begin {gather*} -\frac {\frac {A\,d^3}{2\,b}-\frac {x^3\,\left (-B\,b^4\,e^3+3\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-3\,B\,b\,c^3\,d^3-9\,A\,b\,c^3\,d^2\,e+6\,A\,c^4\,d^3\right )}{b^4\,c}+\frac {x^2\,\left (B\,b^4\,e^3+3\,B\,b^3\,c\,d\,e^2+A\,b^3\,c\,e^3-9\,B\,b^2\,c^2\,d^2\,e-9\,A\,b^2\,c^2\,d\,e^2+9\,B\,b\,c^3\,d^3+27\,A\,b\,c^3\,d^2\,e-18\,A\,c^4\,d^3\right )}{2\,b^3\,c^2}+\frac {d^2\,x\,\left (3\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,d\,\mathrm {atanh}\left (\frac {3\,d\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b\,\left (3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2-3\,B\,b\,c\,d^3-9\,A\,b\,c\,d^2\,e+6\,A\,c^2\,d^3\right )}\right )\,\left (b\,e-c\,d\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 28.93, size = 653, normalized size = 3.53 \begin {gather*} \frac {- A b^{3} c^{2} d^{3} + x^{3} \left (6 A b^{2} c^{3} d e^{2} - 18 A b c^{4} d^{2} e + 12 A c^{5} d^{3} - 2 B b^{4} c e^{3} + 6 B b^{2} c^{3} d^{2} e - 6 B b c^{4} d^{3}\right ) + x^{2} \left (- A b^{4} c e^{3} + 9 A b^{3} c^{2} d e^{2} - 27 A b^{2} c^{3} d^{2} e + 18 A b c^{4} d^{3} - B b^{5} e^{3} - 3 B b^{4} c d e^{2} + 9 B b^{3} c^{2} d^{2} e - 9 B b^{2} c^{3} d^{3}\right ) + x \left (- 6 A b^{3} c^{2} d^{2} e + 4 A b^{2} c^{3} d^{3} - 2 B b^{3} c^{2} d^{3}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac {3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} - 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} - \frac {3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} + 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________