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 = 0.04, antiderivative size = 143, 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} \begin {gather*} -\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} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 99, normalized size = 0.69 \begin {gather*} \frac {c \left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (-6 a^3+6 a^2 b (n+1) x-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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^n}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.15, size = 186, normalized size = 1.30 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} {\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 136, normalized size = 0.95 \begin {gather*} -\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \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} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 116, normalized size = 0.81 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 229, normalized size = 1.60 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {c^2\,x^4\,\sqrt {c\,x^2}\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4\,c^2\,\sqrt {c\,x^2}}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,c^2\,n\,x\,\sqrt {c\,x^2}}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,c^2\,n\,x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,c^2\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{x} \end {gather*}
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 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )^{n}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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