Optimal. Leaf size=169 \[ -\frac{2 b^2 (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5}+\frac{6 b \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}+\frac{2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^3 B (d+e x)^{5/2}}{5 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0666049, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ -\frac{2 b^2 (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5}+\frac{6 b \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}+\frac{2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^3 B (d+e x)^{5/2}}{5 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{5/2}}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{3/2}}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 \sqrt{d+e x}}+\frac{b^2 (-4 b B d+A b e+3 a B e) \sqrt{d+e x}}{e^4}+\frac{b^3 B (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{e^5 \sqrt{d+e x}}+\frac{6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt{d+e x}}{e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{3/2}}{3 e^5}+\frac{2 b^3 B (d+e x)^{5/2}}{5 e^5}\\ \end{align*}
Mathematica [A] time = 0.137473, size = 145, normalized size = 0.86 \[ \frac{2 \left (-5 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+45 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)+15 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)-5 (b d-a e)^3 (B d-A e)+3 b^3 B (d+e x)^4\right )}{15 e^5 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 301, normalized size = 1.8 \begin{align*} -{\frac{-6\,{b}^{3}B{x}^{4}{e}^{4}-10\,A{b}^{3}{e}^{4}{x}^{3}-30\,Ba{b}^{2}{e}^{4}{x}^{3}+16\,B{b}^{3}d{e}^{3}{x}^{3}-90\,Aa{b}^{2}{e}^{4}{x}^{2}+60\,A{b}^{3}d{e}^{3}{x}^{2}-90\,B{a}^{2}b{e}^{4}{x}^{2}+180\,Ba{b}^{2}d{e}^{3}{x}^{2}-96\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+90\,A{a}^{2}b{e}^{4}x-360\,Aa{b}^{2}d{e}^{3}x+240\,A{b}^{3}{d}^{2}{e}^{2}x+30\,B{a}^{3}{e}^{4}x-360\,B{a}^{2}bd{e}^{3}x+720\,Ba{b}^{2}{d}^{2}{e}^{2}x-384\,B{b}^{3}{d}^{3}ex+10\,{a}^{3}A{e}^{4}+60\,A{a}^{2}bd{e}^{3}-240\,Aa{b}^{2}{d}^{2}{e}^{2}+160\,A{b}^{3}{d}^{3}e+20\,B{a}^{3}d{e}^{3}-240\,B{a}^{2}b{d}^{2}{e}^{2}+480\,Ba{b}^{2}{d}^{3}e-256\,B{b}^{3}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02987, size = 366, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{3} - 5 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 45 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\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} - 3 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5329, size = 605, normalized size = 3.58 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} - 5 \, A a^{3} e^{4} - 80 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 120 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (64 \, B b^{3} d^{3} e - 40 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 60 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 58.7335, size = 199, normalized size = 1.18 \begin{align*} \frac{2 B b^{3} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (6 A a b^{2} e^{2} - 6 A b^{3} d e + 6 B a^{2} b e^{2} - 18 B a b^{2} d e + 12 B b^{3} d^{2}\right )}{e^{5}} - \frac{2 \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{e^{5} \sqrt{d + e x}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.48557, size = 493, normalized size = 2.92 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e^{20} + 90 \, \sqrt{x e + d} B b^{3} d^{2} e^{20} + 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{21} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{21} - 135 \, \sqrt{x e + d} B a b^{2} d e^{21} - 45 \, \sqrt{x e + d} A b^{3} d e^{21} + 45 \, \sqrt{x e + d} B a^{2} b e^{22} + 45 \, \sqrt{x e + d} A a b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} B b^{3} d^{3} - B b^{3} d^{4} - 27 \,{\left (x e + d\right )} B a b^{2} d^{2} e - 9 \,{\left (x e + d\right )} A b^{3} d^{2} e + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 18 \,{\left (x e + d\right )} B a^{2} b d e^{2} + 18 \,{\left (x e + d\right )} A a b^{2} d e^{2} - 3 \, B a^{2} b d^{2} e^{2} - 3 \, A a b^{2} d^{2} e^{2} - 3 \,{\left (x e + d\right )} B a^{3} e^{3} - 9 \,{\left (x e + d\right )} A a^{2} b e^{3} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} - A a^{3} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]