Optimal. Leaf size=124 \[ \frac {(b d-a e)^3 (B d-A e) \log (d+e x)}{e^5}-\frac {b x (b d-a e)^2 (B d-A e)}{e^4}+\frac {(a+b x)^2 (b d-a e) (B d-A e)}{2 e^3}-\frac {(a+b x)^3 (B d-A e)}{3 e^2}+\frac {B (a+b x)^4}{4 b e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {(a+b x)^3 (B d-A e)}{3 e^2}+\frac {(a+b x)^2 (b d-a e) (B d-A e)}{2 e^3}-\frac {b x (b d-a e)^2 (B d-A e)}{e^4}+\frac {(b d-a e)^3 (B d-A e) \log (d+e x)}{e^5}+\frac {B (a+b x)^4}{4 b e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{d+e x} \, dx &=\int \left (\frac {b (b d-a e)^2 (-B d+A e)}{e^4}-\frac {b (b d-a e) (-B d+A e) (a+b x)}{e^3}+\frac {b (-B d+A e) (a+b x)^2}{e^2}+\frac {B (a+b x)^3}{e}+\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {b (b d-a e)^2 (B d-A e) x}{e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^2}{2 e^3}-\frac {(B d-A e) (a+b x)^3}{3 e^2}+\frac {B (a+b x)^4}{4 b e}+\frac {(b d-a e)^3 (B d-A e) \log (d+e x)}{e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 169, normalized size = 1.36 \[ \frac {e x \left (12 a^3 B e^3+18 a^2 b e^2 (2 A e-2 B d+B e x)+6 a b^2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )+12 (b d-a e)^3 (B d-A e) \log (d+e x)}{12 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.82, size = 260, normalized size = 2.10 \[ \frac {3 \, B b^{3} e^{4} x^{4} - 4 \, {\left (B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 12 \, {\left (B b^{3} d^{3} e - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.33, size = 284, normalized size = 2.29 \[ {\left (B b^{3} d^{4} - 3 \, B a b^{2} d^{3} e - A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} - B a^{3} d e^{3} - 3 \, A a^{2} b d e^{3} + A a^{3} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B b^{3} x^{4} e^{3} - 4 \, B b^{3} d x^{3} e^{2} + 6 \, B b^{3} d^{2} x^{2} e - 12 \, B b^{3} d^{3} x + 12 \, B a b^{2} x^{3} e^{3} + 4 \, A b^{3} x^{3} e^{3} - 18 \, B a b^{2} d x^{2} e^{2} - 6 \, A b^{3} d x^{2} e^{2} + 36 \, B a b^{2} d^{2} x e + 12 \, A b^{3} d^{2} x e + 18 \, B a^{2} b x^{2} e^{3} + 18 \, A a b^{2} x^{2} e^{3} - 36 \, B a^{2} b d x e^{2} - 36 \, A a b^{2} d x e^{2} + 12 \, B a^{3} x e^{3} + 36 \, A a^{2} b x e^{3}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 341, normalized size = 2.75 \[ \frac {B \,b^{3} x^{4}}{4 e}+\frac {A \,b^{3} x^{3}}{3 e}+\frac {B a \,b^{2} x^{3}}{e}-\frac {B \,b^{3} d \,x^{3}}{3 e^{2}}+\frac {3 A a \,b^{2} x^{2}}{2 e}-\frac {A \,b^{3} d \,x^{2}}{2 e^{2}}+\frac {3 B \,a^{2} b \,x^{2}}{2 e}-\frac {3 B a \,b^{2} d \,x^{2}}{2 e^{2}}+\frac {B \,b^{3} d^{2} x^{2}}{2 e^{3}}+\frac {A \,a^{3} \ln \left (e x +d \right )}{e}-\frac {3 A \,a^{2} b d \ln \left (e x +d \right )}{e^{2}}+\frac {3 A \,a^{2} b x}{e}+\frac {3 A a \,b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 A a \,b^{2} d x}{e^{2}}-\frac {A \,b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {A \,b^{3} d^{2} x}{e^{3}}-\frac {B \,a^{3} d \ln \left (e x +d \right )}{e^{2}}+\frac {B \,a^{3} x}{e}+\frac {3 B \,a^{2} b \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 B \,a^{2} b d x}{e^{2}}-\frac {3 B a \,b^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {3 B a \,b^{2} d^{2} x}{e^{3}}+\frac {B \,b^{3} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {B \,b^{3} d^{3} x}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.50, size = 258, normalized size = 2.08 \[ \frac {3 \, B b^{3} e^{3} x^{4} - 4 \, {\left (B b^{3} d e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (B b^{3} d^{2} e - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{2} - 12 \, {\left (B b^{3} d^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x}{12 \, e^{4}} + \frac {{\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.08, size = 268, normalized size = 2.16 \[ x\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{e}-\frac {B\,b^3\,d}{e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{e}\right )}{e}\right )+x^3\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{3\,e}-\frac {B\,b^3\,d}{3\,e^2}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{e}-\frac {B\,b^3\,d}{e^2}\right )}{2\,e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^3+A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2-3\,A\,a^2\,b\,d\,e^3-3\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2+B\,b^3\,d^4-A\,b^3\,d^3\,e\right )}{e^5}+\frac {B\,b^3\,x^4}{4\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.68, size = 221, normalized size = 1.78 \[ \frac {B b^{3} x^{4}}{4 e} + x^{3} \left (\frac {A b^{3}}{3 e} + \frac {B a b^{2}}{e} - \frac {B b^{3} d}{3 e^{2}}\right ) + x^{2} \left (\frac {3 A a b^{2}}{2 e} - \frac {A b^{3} d}{2 e^{2}} + \frac {3 B a^{2} b}{2 e} - \frac {3 B a b^{2} d}{2 e^{2}} + \frac {B b^{3} d^{2}}{2 e^{3}}\right ) + x \left (\frac {3 A a^{2} b}{e} - \frac {3 A a b^{2} d}{e^{2}} + \frac {A b^{3} d^{2}}{e^{3}} + \frac {B a^{3}}{e} - \frac {3 B a^{2} b d}{e^{2}} + \frac {3 B a b^{2} d^{2}}{e^{3}} - \frac {B b^{3} d^{3}}{e^{4}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________