Optimal. Leaf size=167 \[ -\frac{2 b^2 (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5}+\frac{2 b (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5}-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5} \]
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Rubi [A] time = 0.067525, antiderivative size = 167, 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} \[ -\frac{2 b^2 (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5}+\frac{2 b (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5}-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{3/2}}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 \sqrt{d+e x}}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) \sqrt{d+e x}}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{3/2}}{e^4}+\frac{b^3 B (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt{d+e x}}{e^5}+\frac{2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3/2}}{e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{5/2}}{5 e^5}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.120638, size = 145, normalized size = 0.87 \[ \frac{2 \left (-7 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+35 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-35 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)-35 (b d-a e)^3 (B d-A e)+5 b^3 B (d+e x)^4\right )}{35 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 301, normalized size = 1.8 \begin{align*} -{\frac{-10\,{b}^{3}B{x}^{4}{e}^{4}-14\,A{b}^{3}{e}^{4}{x}^{3}-42\,Ba{b}^{2}{e}^{4}{x}^{3}+16\,B{b}^{3}d{e}^{3}{x}^{3}-70\,Aa{b}^{2}{e}^{4}{x}^{2}+28\,A{b}^{3}d{e}^{3}{x}^{2}-70\,B{a}^{2}b{e}^{4}{x}^{2}+84\,Ba{b}^{2}d{e}^{3}{x}^{2}-32\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}-210\,A{a}^{2}b{e}^{4}x+280\,Aa{b}^{2}d{e}^{3}x-112\,A{b}^{3}{d}^{2}{e}^{2}x-70\,B{a}^{3}{e}^{4}x+280\,B{a}^{2}bd{e}^{3}x-336\,Ba{b}^{2}{d}^{2}{e}^{2}x+128\,B{b}^{3}{d}^{3}ex+70\,{a}^{3}A{e}^{4}-420\,A{a}^{2}bd{e}^{3}+560\,Aa{b}^{2}{d}^{2}{e}^{2}-224\,A{b}^{3}{d}^{3}e-140\,B{a}^{3}d{e}^{3}+560\,B{a}^{2}b{d}^{2}{e}^{2}-672\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{35\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80309, size = 369, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{3} - 7 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\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 )} \sqrt{e x + d}}{e^{4}} - \frac{35 \,{\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 )}}{\sqrt{e x + d} e^{4}}\right )}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31212, size = 583, normalized size = 3.49 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 35 \, A a^{3} e^{4} + 112 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 280 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} +{\left (16 \, B b^{3} d^{2} e^{2} - 14 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 35 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 43.3904, size = 255, normalized size = 1.53 \begin{align*} \frac{2 B b^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \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 )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (6 A a^{2} b e^{3} - 12 A a b^{2} d e^{2} + 6 A b^{3} d^{2} e + 2 B a^{3} e^{3} - 12 B a^{2} b d e^{2} + 18 B a b^{2} d^{2} e - 8 B b^{3} d^{3}\right )}{e^{5}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.95955, size = 514, normalized size = 3.08 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{3} e^{30} - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{30} - 140 \, \sqrt{x e + d} B b^{3} d^{3} e^{30} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{31} + 7 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{31} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{31} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{31} + 315 \, \sqrt{x e + d} B a b^{2} d^{2} e^{31} + 105 \, \sqrt{x e + d} A b^{3} d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{32} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{32} - 210 \, \sqrt{x e + d} B a^{2} b d e^{32} - 210 \, \sqrt{x e + d} A a b^{2} d e^{32} + 35 \, \sqrt{x e + d} B a^{3} e^{33} + 105 \, \sqrt{x e + d} A a^{2} b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\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 )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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