Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]
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Rubi [A] time = 0.0350631, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x) (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac{b B (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{7/2}}{7 e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{9/2}}{9 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3}\\ \end{align*}
Mathematica [A] time = 0.0585851, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{126\,bB{x}^{2}{e}^{2}+154\,Ab{e}^{2}x+154\,Ba{e}^{2}x-56\,Bbdex+198\,aA{e}^{2}-44\,Abde-44\,Bade+16\,bB{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04985, size = 101, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} B b - 77 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82336, size = 435, normalized size = 5.24 \begin{align*} \frac{2 \,{\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \,{\left (B a + A b\right )} d^{4} e + 7 \,{\left (23 \, B b d e^{4} + 11 \,{\left (B a + A b\right )} e^{5}\right )} x^{4} +{\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \,{\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \,{\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \,{\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} -{\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \,{\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.56146, size = 476, normalized size = 5.73 \begin{align*} \begin{cases} \frac{2 A a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 A a d^{2} x \sqrt{d + e x}}{7} + \frac{6 A a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 A a e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 A b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 A b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 A b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 A b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 A b e^{2} x^{4} \sqrt{d + e x}}{9} - \frac{4 B a d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 B a d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 B a d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 B a d e x^{3} \sqrt{d + e x}}{63} + \frac{2 B a e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 B b d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 B b d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 B b d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 B b d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 B b d e x^{4} \sqrt{d + e x}}{99} + \frac{2 B b e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.18995, size = 682, normalized size = 8.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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