Optimal. Leaf size=160 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^3 (a+b x)} \]
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Rubi [A] time = 0.0857776, antiderivative size = 160, 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} \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\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)}{e^2 (d+e x)^{5/2}}+\frac{b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^{3/2}}+\frac{b^2 B}{e^2 \sqrt{d+e x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 (b d-a e) (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac{2 (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}}+\frac{2 b B \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0641243, size = 86, normalized size = 0.54 \[ -\frac{2 \sqrt{(a+b x)^2} \left (a e (A e+2 B d+3 B e x)+A b e (2 d+3 e x)-b B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 88, normalized size = 0.6 \begin{align*} -{\frac{-6\,B{x}^{2}b{e}^{2}+6\,Axb{e}^{2}+6\,aB{e}^{2}x-24\,Bxbde+2\,aA{e}^{2}+4\,Abde+4\,aBde-16\,Bb{d}^{2}}{3\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05356, size = 130, normalized size = 0.81 \begin{align*} -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )} A}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (3 \, b e^{2} x^{2} + 8 \, b d^{2} - 2 \, a d e + 3 \,{\left (4 \, b d e - a e^{2}\right )} x\right )} B}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37365, size = 196, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} - A a e^{2} - 2 \,{\left (B a + A b\right )} d e + 3 \,{\left (4 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \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 [A] time = 1.14827, size = 184, normalized size = 1.15 \begin{align*} 2 \, \sqrt{x e + d} B b e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) + \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d \mathrm{sgn}\left (b x + a\right ) - B b d^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \,{\left (x e + d\right )} B a e \mathrm{sgn}\left (b x + a\right ) - 3 \,{\left (x e + d\right )} A b e \mathrm{sgn}\left (b x + a\right ) + B a d e \mathrm{sgn}\left (b x + a\right ) + A b d e \mathrm{sgn}\left (b x + a\right ) - A a e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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