Optimal. Leaf size=158 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]
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Rubi [A] time = 0.099405, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]
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} \, 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} \, 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}+\frac{b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^4}+\frac{b^2 B}{e^2 (d+e x)^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e) (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{(2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0411271, size = 80, normalized size = 0.51 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 86, normalized size = 0.5 \begin{align*} -{\frac{6\,B{x}^{2}b{e}^{2}+4\,Axb{e}^{2}+4\,aB{e}^{2}x+4\,Bxbde+3\,aA{e}^{2}+Abde+aBde+Bb{d}^{2}}{12\,{e}^{3} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63991, size = 217, normalized size = 1.37 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.92972, size = 117, normalized size = 0.74 \begin{align*} - \frac{3 A a e^{2} + A b d e + B a d e + B b d^{2} + 6 B b e^{2} x^{2} + x \left (4 A b e^{2} + 4 B a e^{2} + 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12921, size = 157, normalized size = 0.99 \begin{align*} -\frac{{\left (6 \, B b x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, B b d x e \mathrm{sgn}\left (b x + a\right ) + B b d^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, B a x e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, A b x e^{2} \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 ) + 3 \, A a e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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