Optimal. Leaf size=78 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^2} \]
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Rubi [A] time = 0.0492313, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {770, 21, 43} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
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 (a+b x) \left (a b+b^2 x\right )^5 (d+e x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x) \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(b d-a e) (a+b x)^6}{b}+\frac{e (a+b x)^7}{b}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^2}+\frac{e (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.0501441, size = 140, normalized size = 1.79 \[ \frac{x \sqrt{(a+b x)^2} \left (70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+56 a^5 b x (3 d+2 e x)+28 a^6 (2 d+e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right )}{56 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 162, normalized size = 2.1 \begin{align*}{\frac{x \left ( 7\,e{b}^{6}{x}^{7}+48\,{x}^{6}ea{b}^{5}+8\,{x}^{6}d{b}^{6}+140\,{x}^{5}e{a}^{2}{b}^{4}+56\,{x}^{5}da{b}^{5}+224\,{a}^{3}{b}^{3}e{x}^{4}+168\,{a}^{2}{b}^{4}d{x}^{4}+210\,{x}^{3}e{a}^{4}{b}^{2}+280\,{x}^{3}d{a}^{3}{b}^{3}+112\,{a}^{5}be{x}^{2}+280\,{a}^{4}{b}^{2}d{x}^{2}+28\,xe{a}^{6}+168\,xd{a}^{5}b+56\,d{a}^{6} \right ) }{56\, \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 = 1.51014, size = 304, normalized size = 3.9 \begin{align*} \frac{1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac{1}{7} \,{\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} +{\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} +{\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d + a^{6} 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 (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 = 1.12021, size = 319, normalized size = 4.09 \begin{align*} \frac{1}{8} \, b^{6} x^{8} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{7} \, a b^{5} x^{7} e \mathrm{sgn}\left (b x + a\right ) + a b^{5} d x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{4} x^{6} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d x^{5} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{3} b^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{4} \, a^{4} b^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d x^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{5} b x^{3} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a^{6} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{6} 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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