Optimal. Leaf size=99 \[ \frac{a^2 x (a+b x)^{n+1}}{b^3 c^2 (n+1) \sqrt{c x^2}}-\frac{2 a x (a+b x)^{n+2}}{b^3 c^2 (n+2) \sqrt{c x^2}}+\frac{x (a+b x)^{n+3}}{b^3 c^2 (n+3) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0307015, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{a^2 x (a+b x)^{n+1}}{b^3 c^2 (n+1) \sqrt{c x^2}}-\frac{2 a x (a+b x)^{n+2}}{b^3 c^2 (n+2) \sqrt{c x^2}}+\frac{x (a+b x)^{n+3}}{b^3 c^2 (n+3) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7 (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int x^2 (a+b x)^n \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2 (a+b x)^n}{b^2}-\frac{2 a (a+b x)^{1+n}}{b^2}+\frac{(a+b x)^{2+n}}{b^2}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{a^2 x (a+b x)^{1+n}}{b^3 c^2 (1+n) \sqrt{c x^2}}-\frac{2 a x (a+b x)^{2+n}}{b^3 c^2 (2+n) \sqrt{c x^2}}+\frac{x (a+b x)^{3+n}}{b^3 c^2 (3+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0300613, size = 70, normalized size = 0.71 \[ \frac{x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 c^2 (n+1) (n+2) (n+3) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 83, normalized size = 0.8 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{n}^{2}{x}^{2}+3\,{b}^{2}n{x}^{2}-2\,abnx+2\,{b}^{2}{x}^{2}-2\,abx+2\,{a}^{2} \right ){x}^{5}}{{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05935, size = 112, normalized size = 1.13 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \sqrt{c} x^{3} +{\left (n^{2} + n\right )} a b^{2} \sqrt{c} x^{2} - 2 \, a^{2} b \sqrt{c} n x + 2 \, a^{3} \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35503, size = 231, normalized size = 2.33 \begin{align*} -\frac{{\left (2 \, a^{2} b n x -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} -{\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{3} c^{3} n^{3} + 6 \, b^{3} c^{3} n^{2} + 11 \, b^{3} c^{3} n + 6 \, b^{3} c^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n} x^{7}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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