Optimal. Leaf size=143 \[ -\frac{a^3 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]
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Rubi [A] time = 0.0424376, antiderivative size = 143, 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} \[ -\frac{a^3 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2} (a+b x)^n}{x^2} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^3 (a+b x)^n \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (-\frac{a^3 (a+b x)^n}{b^3}+\frac{3 a^2 (a+b x)^{1+n}}{b^3}-\frac{3 a (a+b x)^{2+n}}{b^3}+\frac{(a+b x)^{3+n}}{b^3}\right ) \, dx}{x}\\ &=-\frac{a^3 c^2 \sqrt{c x^2} (a+b x)^{1+n}}{b^4 (1+n) x}+\frac{3 a^2 c^2 \sqrt{c x^2} (a+b x)^{2+n}}{b^4 (2+n) x}-\frac{3 a c^2 \sqrt{c x^2} (a+b x)^{3+n}}{b^4 (3+n) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{4+n}}{b^4 (4+n) x}\\ \end{align*}
Mathematica [A] time = 0.0187142, size = 99, normalized size = 0.69 \[ \frac{c \left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (6 a^2 b (n+1) x-6 a^3-3 a b^2 \left (n^2+3 n+2\right ) x^2+b^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )}{b^4 (n+1) (n+2) (n+3) (n+4) x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 136, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{n}^{3}{x}^{3}-6\,{b}^{3}{n}^{2}{x}^{3}+3\,a{b}^{2}{n}^{2}{x}^{2}-11\,{b}^{3}n{x}^{3}+9\,a{b}^{2}n{x}^{2}-6\,{b}^{3}{x}^{3}-6\,{a}^{2}bnx+6\,a{b}^{2}{x}^{2}-6\,{a}^{2}bx+6\,{a}^{3} \right ) }{{x}^{5}{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \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 = 1.04549, size = 157, normalized size = 1.1 \begin{align*} \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{\frac{5}{2}} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{\frac{5}{2}} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} c^{\frac{5}{2}} x^{2} + 6 \, a^{3} b c^{\frac{5}{2}} n x - 6 \, a^{4} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.626, size = 373, normalized size = 2.61 \begin{align*} \frac{{\left (6 \, a^{3} b c^{2} n x - 6 \, a^{4} c^{2} +{\left (b^{4} c^{2} n^{3} + 6 \, b^{4} c^{2} n^{2} + 11 \, b^{4} c^{2} n + 6 \, b^{4} c^{2}\right )} x^{4} +{\left (a b^{3} c^{2} n^{3} + 3 \, a b^{3} c^{2} n^{2} + 2 \, a b^{3} c^{2} n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} c^{2} n^{2} + a^{2} b^{2} c^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )} x} \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 \frac{\left (c x^{2}\right )^{\frac{5}{2}} \left (a + b x\right )^{n}}{x^{2}}\, dx \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 (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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