Optimal. Leaf size=135 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-2 a B e+A b e+b B d)}{7 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)}{6 b^3}+\frac{B e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
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Rubi [A] time = 0.203986, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 77} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-2 a B e+A b e+b B d)}{7 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)}{6 b^3}+\frac{B e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) (d+e x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{(A b-a B) (b d-a e) \left (a b+b^2 x\right )^5}{b^2}+\frac{(b B d+A b e-2 a B e) \left (a b+b^2 x\right )^6}{b^3}+\frac{B e \left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (b d-a e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac{(b B d+A b e-2 a B e) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac{B e (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.11434, size = 214, normalized size = 1.59 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^3 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+28 a^2 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+70 a^4 b x (A (6 d+4 e x)+B x (4 d+3 e x))+28 a^5 (3 A (2 d+e x)+B x (3 d+2 e x))+4 a b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))\right )}{168 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 284, normalized size = 2.1 \begin{align*}{\frac{x \left ( 21\,Be{b}^{5}{x}^{7}+24\,{x}^{6}A{b}^{5}e+120\,{x}^{6}Bea{b}^{4}+24\,{x}^{6}B{b}^{5}d+140\,{x}^{5}Aa{b}^{4}e+28\,{x}^{5}Ad{b}^{5}+280\,{x}^{5}Be{a}^{2}{b}^{3}+140\,{x}^{5}Ba{b}^{4}d+336\,A{a}^{2}{b}^{3}e{x}^{4}+168\,Aa{b}^{4}d{x}^{4}+336\,B{a}^{3}{b}^{2}e{x}^{4}+336\,B{a}^{2}{b}^{3}d{x}^{4}+420\,{x}^{3}A{a}^{3}{b}^{2}e+420\,{x}^{3}Ad{a}^{2}{b}^{3}+210\,{x}^{3}Be{a}^{4}b+420\,{x}^{3}B{a}^{3}{b}^{2}d+280\,{x}^{2}A{a}^{4}be+560\,{x}^{2}Ad{a}^{3}{b}^{2}+56\,{x}^{2}Be{a}^{5}+280\,{x}^{2}B{a}^{4}bd+84\,xA{a}^{5}e+420\,xAd{a}^{4}b+84\,xB{a}^{5}d+168\,Ad{a}^{5} \right ) }{168\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 [B] time = 1.54318, size = 509, normalized size = 3.77 \begin{align*} \frac{1}{8} \, B b^{5} e x^{8} + A a^{5} d x + \frac{1}{7} \,{\left (B b^{5} d +{\left (5 \, B a b^{4} + A b^{5}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e\right )} x^{6} +{\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e\right )} x^{5} + \frac{5}{4} \,{\left (2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d +{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d +{\left (B a^{5} + 5 \, A a^{4} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{5} e +{\left (B a^{5} + 5 \, A a^{4} b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14919, size = 574, normalized size = 4.25 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, B b^{5} d x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, B a b^{4} x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, A b^{5} x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, B a b^{4} d x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A b^{5} d x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, B a^{2} b^{3} x^{6} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, A a b^{4} x^{6} e \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d x^{5} \mathrm{sgn}\left (b x + a\right ) + A a b^{4} d x^{5} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{3} b^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{2} b^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, B a^{3} b^{2} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{2} b^{3} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, B a^{4} b x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{3} b^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, B a^{4} b d x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, A a^{3} b^{2} d x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a^{5} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, A a^{4} b x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a^{5} d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{4} b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a^{5} x^{2} e \mathrm{sgn}\left (b x + a\right ) + A a^{5} d x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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