Optimal. Leaf size=156 \[ \frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac {A d^4}{b^2 x}+\frac {(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 156, 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 {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac {(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}-\frac {A d^4}{b^2 x}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {e^3 (4 B c d-2 b B e+A c e)}{c^3}+\frac {A d^4}{b^2 x^2}+\frac {d^3 (b B d-2 A c d+4 A b e)}{b^3 x}+\frac {B e^4 x}{c^2}-\frac {(b B-A c) (-c d+b e)^4}{b^2 c^3 (b+c x)^2}+\frac {(c d-b e)^3 \left (2 A c^2 d-3 b^2 B e-b c (B d-2 A e)\right )}{b^3 c^3 (b+c x)}\right ) \, dx\\ &=-\frac {A d^4}{b^2 x}+\frac {e^3 (4 B c d-2 b B e+A c e) x}{c^3}+\frac {B e^4 x^2}{2 c^2}+\frac {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac {d^3 (b B d-2 A c d+4 A b e) \log (x)}{b^3}+\frac {(c d-b e)^3 \left (2 A c^2 d-3 b^2 B e-b c (B d-2 A e)\right ) \log (b+c x)}{b^3 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 155, normalized size = 0.99 \begin {gather*} \frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac {A d^4}{b^2 x}+\frac {(b e-c d)^3 \log (b+c x) \left (b c (B d-2 A e)-2 A c^2 d+3 b^2 B e\right )}{b^3 c^4}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \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)^4}{\left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 517, normalized size = 3.31 \begin {gather*} \frac {B b^{3} c^{3} e^{4} x^{4} - 2 \, A b^{2} c^{4} d^{4} + {\left (8 \, B b^{3} c^{3} d e^{3} - {\left (3 \, B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} e^{4}\right )} x^{3} + 2 \, {\left (4 \, B b^{4} c^{2} d e^{3} - {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + 2 \, {\left ({\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 4 \, {\left (B b^{3} c^{3} - A b^{2} c^{4}\right )} d^{3} e + 6 \, {\left (B b^{4} c^{2} - A b^{3} c^{3}\right )} d^{2} e^{2} - 4 \, {\left (B b^{5} c - A b^{4} c^{2}\right )} d e^{3} + {\left (B b^{6} - A b^{5} c\right )} e^{4}\right )} x - 2 \, {\left ({\left (4 \, A b c^{5} d^{3} e - 6 \, B b^{3} c^{3} d^{2} e^{2} + {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \, {\left (2 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} d e^{3} - {\left (3 \, B b^{5} c - 2 \, A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + {\left (4 \, A b^{2} c^{4} d^{3} e - 6 \, B b^{4} c^{2} d^{2} e^{2} + {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} + 4 \, {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} d e^{3} - {\left (3 \, B b^{6} - 2 \, A b^{5} c\right )} e^{4}\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (4 \, A b c^{5} d^{3} e + {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4}\right )} x^{2} + {\left (4 \, A b^{2} c^{4} d^{3} e + {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4}\right )} x\right )} \log \relax (x)}{2 \, {\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 315, normalized size = 2.02 \begin {gather*} \frac {{\left (B b d^{4} - 2 \, A c d^{4} + 4 \, A b d^{3} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} - 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac {{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} + 4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + 8 \, B b^{4} c d e^{3} - 4 \, A b^{3} c^{2} d e^{3} - 3 \, B b^{5} e^{4} + 2 \, A b^{4} c e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac {A b c^{4} d^{4} - {\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 403, normalized size = 2.58 \begin {gather*} \frac {B \,e^{4} x^{2}}{2 c^{2}}-\frac {A \,b^{2} e^{4}}{\left (c x +b \right ) c^{3}}+\frac {4 A b d \,e^{3}}{\left (c x +b \right ) c^{2}}-\frac {2 A b \,e^{4} \ln \left (c x +b \right )}{c^{3}}+\frac {4 A \,d^{3} e}{\left (c x +b \right ) b}-\frac {A c \,d^{4}}{\left (c x +b \right ) b^{2}}+\frac {4 A \,d^{3} e \ln \relax (x )}{b^{2}}-\frac {4 A \,d^{3} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 A c \,d^{4} \ln \relax (x )}{b^{3}}+\frac {2 A c \,d^{4} \ln \left (c x +b \right )}{b^{3}}-\frac {6 A \,d^{2} e^{2}}{\left (c x +b \right ) c}+\frac {4 A d \,e^{3} \ln \left (c x +b \right )}{c^{2}}+\frac {A \,e^{4} x}{c^{2}}+\frac {B \,b^{3} e^{4}}{\left (c x +b \right ) c^{4}}-\frac {4 B \,b^{2} d \,e^{3}}{\left (c x +b \right ) c^{3}}+\frac {3 B \,b^{2} e^{4} \ln \left (c x +b \right )}{c^{4}}+\frac {6 B b \,d^{2} e^{2}}{\left (c x +b \right ) c^{2}}-\frac {8 B b d \,e^{3} \ln \left (c x +b \right )}{c^{3}}-\frac {2 B b \,e^{4} x}{c^{3}}+\frac {B \,d^{4}}{\left (c x +b \right ) b}+\frac {B \,d^{4} \ln \relax (x )}{b^{2}}-\frac {B \,d^{4} \ln \left (c x +b \right )}{b^{2}}-\frac {4 B \,d^{3} e}{\left (c x +b \right ) c}+\frac {6 B \,d^{2} e^{2} \ln \left (c x +b \right )}{c^{2}}+\frac {4 B d \,e^{3} x}{c^{2}}-\frac {A \,d^{4}}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.50, size = 310, normalized size = 1.99 \begin {gather*} -\frac {A b c^{4} d^{4} - {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} - 4 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \, {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \, {\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} + {\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} + \frac {{\left (4 \, A b d^{3} e + {\left (B b - 2 \, A c\right )} d^{4}\right )} \log \relax (x)}{b^{3}} + \frac {B c e^{4} x^{2} + 2 \, {\left (4 \, B c d e^{3} - {\left (2 \, B b - A c\right )} e^{4}\right )} x}{2 \, c^{3}} - \frac {{\left (4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \, {\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} d e^{3} - {\left (3 \, B b^{5} - 2 \, A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.74, size = 331, normalized size = 2.12 \begin {gather*} \ln \left (b+c\,x\right )\,\left (\frac {c^2\,\left (6\,B\,b^3\,d^2\,e^2+4\,A\,b^3\,d\,e^3\right )-c\,\left (2\,A\,b^4\,e^4+8\,B\,d\,b^4\,e^3\right )+3\,B\,b^5\,e^4}{b^3\,c^4}-\frac {B\,b\,d^4+4\,A\,b\,e\,d^3}{b^3}+\frac {2\,A\,c\,d^4}{b^3}\right )-\frac {\frac {A\,c^3\,d^4}{b}+\frac {x\,\left (-B\,b^5\,e^4+4\,B\,b^4\,c\,d\,e^3+A\,b^4\,c\,e^4-6\,B\,b^3\,c^2\,d^2\,e^2-4\,A\,b^3\,c^2\,d\,e^3+4\,B\,b^2\,c^3\,d^3\,e+6\,A\,b^2\,c^3\,d^2\,e^2-B\,b\,c^4\,d^4-4\,A\,b\,c^4\,d^3\,e+2\,A\,c^5\,d^4\right )}{b^2\,c}}{c^4\,x^2+b\,c^3\,x}+x\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{c^2}-\frac {2\,B\,b\,e^4}{c^3}\right )+\frac {\ln \relax (x)\,\left (b\,\left (B\,d^4+4\,A\,e\,d^3\right )-2\,A\,c\,d^4\right )}{b^3}+\frac {B\,e^4\,x^2}{2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 15.98, size = 644, normalized size = 4.13 \begin {gather*} \frac {B e^{4} x^{2}}{2 c^{2}} + x \left (\frac {A e^{4}}{c^{2}} - \frac {2 B b e^{4}}{c^{3}} + \frac {4 B d e^{3}}{c^{2}}\right ) + \frac {- A b c^{4} d^{4} + x \left (- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} + \frac {d^{3} \left (4 A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + b c^{3} d^{3} \left (4 A b e - 2 A c d + B b d\right )}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3}} + \frac {\left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right ) \log {\left (x + \frac {- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + \frac {b \left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right )}{c}}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________