Optimal. Leaf size=89 \[ \frac {a^3 c \sqrt {c x^2}}{b^4 x (a+b x)}+\frac {3 a^2 c \sqrt {c x^2} \log (a+b x)}{b^4 x}-\frac {2 a c \sqrt {c x^2}}{b^3}+\frac {c x \sqrt {c x^2}}{2 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \begin {gather*} \frac {a^3 c \sqrt {c x^2}}{b^4 x (a+b x)}+\frac {3 a^2 c \sqrt {c x^2} \log (a+b x)}{b^4 x}-\frac {2 a c \sqrt {c x^2}}{b^3}+\frac {c x \sqrt {c x^2}}{2 b^2} \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 )^{3/2}}{(a+b x)^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {x^3}{(a+b x)^2} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {2 a c \sqrt {c x^2}}{b^3}+\frac {c x \sqrt {c x^2}}{2 b^2}+\frac {a^3 c \sqrt {c x^2}}{b^4 x (a+b x)}+\frac {3 a^2 c \sqrt {c x^2} \log (a+b x)}{b^4 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 0.80 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \left (2 a^3-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 74, normalized size = 0.83 \begin {gather*} \left (c x^2\right )^{3/2} \left (\frac {3 a^2 \log (a+b x)}{b^4 x^3}+\frac {2 a^3-4 a^2 b x-3 a b^2 x^2+b^3 x^3}{2 b^4 x^3 (a+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 79, normalized size = 0.89 \begin {gather*} \frac {{\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} - 4 \, a^{2} b c x + 2 \, a^{3} c + 6 \, {\left (a^{2} b c x + a^{3} c\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (b^{5} x^{2} + a b^{4} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 80, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, c^{\frac {3}{2}} {\left (\frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{4}} + \frac {2 \, a^{3} \mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b^{4}} - \frac {2 \, {\left (3 \, a^{2} \log \left ({\left | a \right |}\right ) + a^{2}\right )} \mathrm {sgn}\relax (x)}{b^{4}} + \frac {b^{2} x^{2} \mathrm {sgn}\relax (x) - 4 \, a b x \mathrm {sgn}\relax (x)}{b^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 0.85 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{3}+6 a^{2} b x \ln \left (b x +a \right )-3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-4 a^{2} b x +2 a^{3}\right )}{2 \left (b x +a \right ) b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 115, normalized size = 1.29 \begin {gather*} \frac {3 \, \left (-1\right )^{\frac {2 \, c x}{b}} a^{2} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{4}} + \frac {3 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4}} - \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{b^{2} x + a b} + \frac {3 \, \sqrt {c x^{2}} c x}{2 \, b^{2}} - \frac {3 \, \sqrt {c x^{2}} a c}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{{\left (a+b\,x\right )}^2} \,d 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 {3}{2}}}{\left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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