Optimal. Leaf size=137 \[ -\frac {3 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{5/2}}+\frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac {b \sqrt {a x^2+b x^3}}{8 x^4} \]
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Rubi [A] time = 0.18, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2008, 206} \begin {gather*} \frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {3 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{5/2}}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx &=-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}+\frac {1}{8} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^5} \, dx\\ &=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}+\frac {1}{16} b^2 \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx\\ &=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac {\left (3 b^3\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{64 a}\\ &=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}+\frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}+\frac {\left (3 b^4\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{128 a^2}\\ &=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}+\frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{64 a^2}\\ &=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}+\frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac {3 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.31 \begin {gather*} -\frac {2 b^4 \left (x^2 (a+b x)\right )^{5/2} \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {b x}{a}+1\right )}{5 a^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 16.00, size = 109, normalized size = 0.80 \begin {gather*} \frac {\left (x^2 (a+b x)\right )^{3/2} \left (\frac {\sqrt {a+b x} \left (3 a^3-11 a^2 (a+b x)-11 a (a+b x)^2+3 (a+b x)^3\right )}{64 a^2 x^4}-\frac {3 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{5/2}}\right )}{x^3 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 197, normalized size = 1.44 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{4} x^{5} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}, \frac {3 \, \sqrt {-a} b^{4} x^{5} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{64 \, a^{3} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 109, normalized size = 0.80 \begin {gather*} \frac {\frac {3 \, b^{5} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{5} \mathrm {sgn}\relax (x) - 11 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{5} \mathrm {sgn}\relax (x) - 11 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{5} \mathrm {sgn}\relax (x) + 3 \, \sqrt {b x + a} a^{3} b^{5} \mathrm {sgn}\relax (x)}{a^{2} b^{4} x^{4}}}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 0.74 \begin {gather*} \frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-3 a^{2} b^{4} x^{4} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+3 \sqrt {b x +a}\, a^{\frac {11}{2}}-11 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}}-11 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {7}{2}}+3 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {5}{2}}\right )}{64 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \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 (b\,x^3+a\,x^2\right )}^{3/2}}{x^8} \,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 (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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