Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.057379, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^{3/2}}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^3}+\frac{b^2 B (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4}\\ \end{align*}
Mathematica [A] time = 0.114426, size = 107, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (-385 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+495 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-693 (b d-a e)^2 (B d-A e)+315 b^2 B (d+e x)^3\right )}{3465 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 169, normalized size = 1.3 \begin{align*}{\frac{630\,{b}^{2}B{x}^{3}{e}^{3}+770\,A{b}^{2}{e}^{3}{x}^{2}+1540\,Bab{e}^{3}{x}^{2}-420\,B{b}^{2}d{e}^{2}{x}^{2}+1980\,Axab{e}^{3}-440\,Ax{b}^{2}d{e}^{2}+990\,Bx{a}^{2}{e}^{3}-880\,Bxabd{e}^{2}+240\,B{b}^{2}{d}^{2}ex+1386\,A{a}^{2}{e}^{3}-792\,Aabd{e}^{2}+176\,A{b}^{2}{d}^{2}e-396\,B{a}^{2}d{e}^{2}+352\,Bab{d}^{2}e-96\,{b}^{2}B{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.997212, size = 215, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{2} - 385 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.37179, size = 652, normalized size = 5.09 \begin{align*} \frac{2 \,{\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \,{\left (12 \, B b^{2} d e^{4} + 11 \,{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B b^{2} d^{2} e^{3} + 110 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \,{\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} +{\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.5597, size = 586, normalized size = 4.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15307, size = 698, normalized size = 5.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]