Optimal. Leaf size=77 \[ -\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}+\frac {2 b B d-A b e-a B e}{3 e^3 (d+e x)^3}-\frac {b B}{2 e^3 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78}
\begin {gather*} \frac {-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac {b B}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{(d+e x)^5} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^5}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^4}+\frac {b B}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}+\frac {2 b B d-A b e-a B e}{3 e^3 (d+e x)^3}-\frac {b B}{2 e^3 (d+e x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 62, normalized size = 0.81 \begin {gather*} -\frac {a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 79, normalized size = 1.03
method | result | size |
risch | \(\frac {-\frac {B b \,x^{2}}{2 e}-\frac {\left (A b e +B a e +B b d \right ) x}{3 e^{2}}-\frac {3 A a \,e^{2}+A b d e +B a d e +B b \,d^{2}}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(68\) |
gosper | \(-\frac {6 b B \,x^{2} e^{2}+4 A b \,e^{2} x +4 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}+A b d e +B a d e +B b \,d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(70\) |
norman | \(\frac {-\frac {B b \,x^{2}}{2 e}-\frac {\left (A b \,e^{2}+B a \,e^{2}+B b d e \right ) x}{3 e^{3}}-\frac {3 A a \,e^{3}+A b d \,e^{2}+B a d \,e^{2}+B b \,d^{2} e}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(78\) |
default | \(-\frac {b B}{2 e^{3} \left (e x +d \right )^{2}}-\frac {A a \,e^{2}-A b d e -B a d e +B b \,d^{2}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {A b e +B a e -2 B b d}{3 e^{3} \left (e x +d \right )^{3}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 98, normalized size = 1.27 \begin {gather*} -\frac {6 \, B b x^{2} e^{2} + B b d^{2} + 3 \, A a e^{2} + {\left (B a e + A b e\right )} d + 4 \, {\left (B b d e + B a e^{2} + A b e^{2}\right )} x}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 94, normalized size = 1.22 \begin {gather*} -\frac {B b d^{2} + {\left (6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x\right )} e^{2} + {\left (4 \, B b d x + {\left (B a + A b\right )} d\right )} e}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.59, size = 117, normalized size = 1.52 \begin {gather*} \frac {- 3 A a e^{2} - A b d e - B a d e - B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 4 A b e^{2} - 4 B a e^{2} - 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.26, size = 121, normalized size = 1.57 \begin {gather*} -\frac {1}{12} \, {\left (\frac {6 \, B b e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {8 \, B b d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, B b d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} + \frac {4 \, A b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} - \frac {3 \, A b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.06, size = 101, normalized size = 1.31 \begin {gather*} -\frac {\frac {3\,A\,a\,e^2+B\,b\,d^2+A\,b\,d\,e+B\,a\,d\,e}{12\,e^3}+\frac {x\,\left (A\,b\,e+B\,a\,e+B\,b\,d\right )}{3\,e^2}+\frac {B\,b\,x^2}{2\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________