Optimal. Leaf size=164 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)} \]
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Rubi [A] time = 0.0821467, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (A+B x) (d+e x)^{5/2} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac{b (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac{b^2 B (d+e x)^{9/2}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}+\frac{2 b B (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.079309, size = 88, normalized size = 0.54 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 89, normalized size = 0.5 \begin{align*}{\frac{126\,B{x}^{2}b{e}^{2}+154\,Axb{e}^{2}+154\,aB{e}^{2}x-56\,Bxbde+198\,aA{e}^{2}-44\,Abde-44\,aBde+16\,Bb{d}^{2}}{693\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03834, size = 289, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e +{\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \,{\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} +{\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x + d} A}{63 \, e^{2}} + \frac{2 \,{\left (63 \, b e^{5} x^{5} + 8 \, b d^{5} - 22 \, a d^{4} e + 7 \,{\left (23 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} +{\left (113 \, b d^{2} e^{3} + 209 \, a d e^{4}\right )} x^{3} + 3 \,{\left (b d^{3} e^{2} + 55 \, a d^{2} e^{3}\right )} x^{2} -{\left (4 \, b d^{4} e - 11 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} B}{693 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29877, size = 435, normalized size = 2.65 \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21454, size = 779, normalized size = 4.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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