Optimal. Leaf size=207 \[ \frac {e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac {e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac {(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac {e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac {A d^4 \log (x)}{b}+\frac {B e^4 x^4}{4 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 207, 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 {e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac {e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac {e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac {(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac {A d^4 \log (x)}{b}+\frac {B e^4 x^4}{4 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^4}{b x+c x^2} \, dx &=\int \left (\frac {e \left (A c e \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{c^4}+\frac {A d^4}{b x}+\frac {e^2 \left (A c e (4 c d-b e)+B \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )\right ) x}{c^3}+\frac {e^3 (4 B c d-b B e+A c e) x^2}{c^2}+\frac {B e^4 x^3}{c}+\frac {(b B-A c) (-c d+b e)^4}{b c^4 (b+c x)}\right ) \, dx\\ &=\frac {e \left (A c e \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{c^4}+\frac {e^2 \left (A c e (4 c d-b e)+B \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )\right ) x^2}{2 c^3}+\frac {e^3 (4 B c d-b B e+A c e) x^3}{3 c^2}+\frac {B e^4 x^4}{4 c}+\frac {A d^4 \log (x)}{b}+\frac {(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 187, normalized size = 0.90 \begin {gather*} \frac {e x \left (2 A c e \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+B \left (-12 b^3 e^3+6 b^2 c e^2 (8 d+e x)-4 b c^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )\right )}{12 c^4}+\frac {(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac {A d^4 \log (x)}{b} \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}{b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 331, normalized size = 1.60 \begin {gather*} \frac {3 \, B b c^{4} e^{4} x^{4} + 12 \, A c^{5} d^{4} \log \relax (x) + 4 \, {\left (4 \, B b c^{4} d e^{3} - {\left (B b^{2} c^{3} - A b c^{4}\right )} e^{4}\right )} x^{3} + 6 \, {\left (6 \, B b c^{4} d^{2} e^{2} - 4 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d e^{3} + {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B b c^{4} d^{3} e - 6 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{3} - {\left (B b^{4} c - A b^{3} c^{2}\right )} e^{4}\right )} x + 12 \, {\left ({\left (B b c^{4} - 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 )} \log \left (c x + b\right )}{12 \, b c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 330, normalized size = 1.59 \begin {gather*} \frac {A d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac {3 \, B c^{3} x^{4} e^{4} + 16 \, B c^{3} d x^{3} e^{3} + 36 \, B c^{3} d^{2} x^{2} e^{2} + 48 \, B c^{3} d^{3} x e - 4 \, B b c^{2} x^{3} e^{4} + 4 \, A c^{3} x^{3} e^{4} - 24 \, B b c^{2} d x^{2} e^{3} + 24 \, A c^{3} d x^{2} e^{3} - 72 \, B b c^{2} d^{2} x e^{2} + 72 \, A c^{3} d^{2} x e^{2} + 6 \, B b^{2} c x^{2} e^{4} - 6 \, A b c^{2} x^{2} e^{4} + 48 \, B b^{2} c d x e^{3} - 48 \, A b c^{2} d x e^{3} - 12 \, B b^{3} x e^{4} + 12 \, A b^{2} c x e^{4}}{12 \, c^{4}} + \frac {{\left (B b c^{4} d^{4} - 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 )} \log \left ({\left | c x + b \right |}\right )}{b c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 396, normalized size = 1.91 \begin {gather*} \frac {B \,e^{4} x^{4}}{4 c}+\frac {A \,e^{4} x^{3}}{3 c}-\frac {B b \,e^{4} x^{3}}{3 c^{2}}+\frac {4 B d \,e^{3} x^{3}}{3 c}-\frac {A b \,e^{4} x^{2}}{2 c^{2}}+\frac {2 A d \,e^{3} x^{2}}{c}+\frac {B \,b^{2} e^{4} x^{2}}{2 c^{3}}-\frac {2 B b d \,e^{3} x^{2}}{c^{2}}+\frac {3 B \,d^{2} e^{2} x^{2}}{c}-\frac {A \,b^{3} e^{4} \ln \left (c x +b \right )}{c^{4}}+\frac {4 A \,b^{2} d \,e^{3} \ln \left (c x +b \right )}{c^{3}}+\frac {A \,b^{2} e^{4} x}{c^{3}}-\frac {6 A b \,d^{2} e^{2} \ln \left (c x +b \right )}{c^{2}}-\frac {4 A b d \,e^{3} x}{c^{2}}+\frac {A \,d^{4} \ln \relax (x )}{b}-\frac {A \,d^{4} \ln \left (c x +b \right )}{b}+\frac {4 A \,d^{3} e \ln \left (c x +b \right )}{c}+\frac {6 A \,d^{2} e^{2} x}{c}+\frac {B \,b^{4} e^{4} \ln \left (c x +b \right )}{c^{5}}-\frac {4 B \,b^{3} d \,e^{3} \ln \left (c x +b \right )}{c^{4}}-\frac {B \,b^{3} e^{4} x}{c^{4}}+\frac {6 B \,b^{2} d^{2} e^{2} \ln \left (c x +b \right )}{c^{3}}+\frac {4 B \,b^{2} d \,e^{3} x}{c^{3}}-\frac {4 B b \,d^{3} e \ln \left (c x +b \right )}{c^{2}}-\frac {6 B b \,d^{2} e^{2} x}{c^{2}}+\frac {B \,d^{4} \ln \left (c x +b \right )}{c}+\frac {4 B \,d^{3} e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 309, normalized size = 1.49 \begin {gather*} \frac {A d^{4} \log \relax (x)}{b} + \frac {3 \, B c^{3} e^{4} x^{4} + 4 \, {\left (4 \, B c^{3} d e^{3} - {\left (B b c^{2} - A c^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (6 \, B c^{3} d^{2} e^{2} - 4 \, {\left (B b c^{2} - A c^{3}\right )} d e^{3} + {\left (B b^{2} c - A b c^{2}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B c^{3} d^{3} e - 6 \, {\left (B b c^{2} - A c^{3}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} c - A b c^{2}\right )} d e^{3} - {\left (B b^{3} - A b^{2} c\right )} e^{4}\right )} x}{12 \, c^{4}} + \frac {{\left ({\left (B b c^{4} - 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 )} \log \left (c x + b\right )}{b c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.67, size = 322, normalized size = 1.56 \begin {gather*} x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{c}-\frac {B\,b\,e^4}{c^2}\right )}{c}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{c}\right )}{c}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{c}-\frac {B\,b\,e^4}{c^2}\right )}{2\,c}-\frac {d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{c}\right )+x^3\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{3\,c}-\frac {B\,b\,e^4}{3\,c^2}\right )-\ln \left (b+c\,x\right )\,\left (\frac {A\,d^4}{b}-\frac {c^4\,\left (B\,b\,d^4+4\,A\,b\,e\,d^3\right )-c\,\left (A\,b^4\,e^4+4\,B\,d\,b^4\,e^3\right )-c^3\,\left (4\,B\,b^2\,d^3\,e+6\,A\,b^2\,d^2\,e^2\right )+c^2\,\left (6\,B\,b^3\,d^2\,e^2+4\,A\,b^3\,d\,e^3\right )+B\,b^5\,e^4}{b\,c^5}\right )+\frac {A\,d^4\,\ln \relax (x)}{b}+\frac {B\,e^4\,x^4}{4\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.18, size = 396, normalized size = 1.91 \begin {gather*} \frac {A d^{4} \log {\relax (x )}}{b} + \frac {B e^{4} x^{4}}{4 c} + x^{3} \left (\frac {A e^{4}}{3 c} - \frac {B b e^{4}}{3 c^{2}} + \frac {4 B d e^{3}}{3 c}\right ) + x^{2} \left (- \frac {A b e^{4}}{2 c^{2}} + \frac {2 A d e^{3}}{c} + \frac {B b^{2} e^{4}}{2 c^{3}} - \frac {2 B b d e^{3}}{c^{2}} + \frac {3 B d^{2} e^{2}}{c}\right ) + x \left (\frac {A b^{2} e^{4}}{c^{3}} - \frac {4 A b d e^{3}}{c^{2}} + \frac {6 A d^{2} e^{2}}{c} - \frac {B b^{3} e^{4}}{c^{4}} + \frac {4 B b^{2} d e^{3}}{c^{3}} - \frac {6 B b d^{2} e^{2}}{c^{2}} + \frac {4 B d^{3} e}{c}\right ) + \frac {\left (- A c + B b\right ) \left (b e - c d\right )^{4} \log {\left (x + \frac {- A b c^{4} d^{4} + \frac {b \left (- A c + B b\right ) \left (b e - c d\right )^{4}}{c}}{- 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 c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________