Optimal. Leaf size=105 \[ \frac {a^2 c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac {2 a c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]
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Rubi [A] time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {a^2 c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac {2 a c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]
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^3} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^2 (a+b x)^n \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \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}{x}\\ &=\frac {a^2 c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^3 (1+n) x}-\frac {2 a c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^3 (2+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{3+n}}{b^3 (3+n) x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.67 \[ \frac {c^3 x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 127, normalized size = 1.21 \[ -\frac {{\left (2 \, a^{2} b c^{2} n x - 2 \, a^{3} c^{2} - {\left (b^{3} c^{2} n^{2} + 3 \, b^{3} c^{2} n + 2 \, b^{3} c^{2}\right )} x^{3} - {\left (a b^{2} c^{2} n^{2} + a b^{2} c^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} {\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.79 \[ \frac {\left (b^{2} n^{2} x^{2}+3 b^{2} n \,x^{2}-2 a b n x +2 b^{2} x^{2}-2 a b x +2 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x +a \right )^{n +1}}{\left (n^{3}+6 n^{2}+11 n +6\right ) b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 80, normalized size = 0.76 \[ \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{\frac {5}{2}} x^{3} + {\left (n^{2} + n\right )} a b^{2} c^{\frac {5}{2}} x^{2} - 2 \, a^{2} b c^{\frac {5}{2}} n x + 2 \, a^{3} c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 154, normalized size = 1.47 \[ \frac {{\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,c^2\,\sqrt {c\,x^2}}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {c^2\,x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}-\frac {2\,a^2\,c^2\,n\,x\,\sqrt {c\,x^2}}{b^2\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,c^2\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )^{n}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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