Optimal. Leaf size=125 \[ \frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{6 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
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Rubi [A] time = 0.139914, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{6 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \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 (d+e x)^2 \, 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{(b d-a e)^2 \left (a b+b^2 x\right )^5}{b^2}+\frac{2 e (b d-a e) \left (a b+b^2 x\right )^6}{b^3}+\frac{e^2 \left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^2 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac{2 e (b d-a e) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac{e^2 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0676425, size = 187, normalized size = 1.5 \[ \frac{x \sqrt{(a+b x)^2} \left (56 a^3 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+28 a^2 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+70 a^4 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+56 a^5 \left (3 d^2+3 d e x+e^2 x^2\right )+8 a b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )}{168 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.153, size = 230, normalized size = 1.8 \begin{align*}{\frac{x \left ( 21\,{e}^{2}{b}^{5}{x}^{7}+120\,{x}^{6}{e}^{2}a{b}^{4}+48\,{x}^{6}de{b}^{5}+280\,{x}^{5}{e}^{2}{a}^{2}{b}^{3}+280\,{x}^{5}dea{b}^{4}+28\,{x}^{5}{d}^{2}{b}^{5}+336\,{a}^{3}{b}^{2}{e}^{2}{x}^{4}+672\,{a}^{2}{b}^{3}de{x}^{4}+168\,a{b}^{4}{d}^{2}{x}^{4}+210\,{x}^{3}{e}^{2}{a}^{4}b+840\,{x}^{3}de{a}^{3}{b}^{2}+420\,{x}^{3}{d}^{2}{a}^{2}{b}^{3}+56\,{x}^{2}{e}^{2}{a}^{5}+560\,{x}^{2}de{a}^{4}b+560\,{x}^{2}{d}^{2}{a}^{3}{b}^{2}+168\,xde{a}^{5}+420\,x{d}^{2}{a}^{4}b+168\,{d}^{2}{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 = 1.59418, size = 417, normalized size = 3.34 \begin{align*} \frac{1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac{1}{7} \,{\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} +{\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\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 (d + e x\right )^{2} \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 = 1.14295, size = 432, normalized size = 3.46 \begin{align*} \frac{1}{8} \, b^{5} x^{8} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{7} \, b^{5} d x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, b^{5} d^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, a b^{4} x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a b^{4} d x^{6} e \mathrm{sgn}\left (b x + a\right ) + a b^{4} d^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{2} b^{3} d x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} d^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b^{2} d x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} d^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, a^{4} b x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{4} b d x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a^{5} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + a^{5} d x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{5} d^{2} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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