Optimal. Leaf size=125 \[ -\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{8 b^{7/2}}+\frac {5 a^2 \sqrt {a x^2+b x^3}}{8 b^3 \sqrt {x}}-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b} \]
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Rubi [A] time = 0.17, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2024, 2029, 206} \begin {gather*} \frac {5 a^2 \sqrt {a x^2+b x^3}}{8 b^3 \sqrt {x}}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{8 b^{7/2}}-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\sqrt {a x^2+b x^3}} \, dx &=\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b}-\frac {(5 a) \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx}{6 b}\\ &=-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b}+\frac {\left (5 a^2\right ) \int \frac {x^{3/2}}{\sqrt {a x^2+b x^3}} \, dx}{8 b^2}\\ &=\frac {5 a^2 \sqrt {a x^2+b x^3}}{8 b^3 \sqrt {x}}-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b}-\frac {\left (5 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^3}} \, dx}{16 b^3}\\ &=\frac {5 a^2 \sqrt {a x^2+b x^3}}{8 b^3 \sqrt {x}}-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b}-\frac {\left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{8 b^3}\\ &=\frac {5 a^2 \sqrt {a x^2+b x^3}}{8 b^3 \sqrt {x}}-\frac {5 a \sqrt {x} \sqrt {a x^2+b x^3}}{12 b^2}+\frac {x^{3/2} \sqrt {a x^2+b x^3}}{3 b}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 104, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x^2 (a+b x)} \left (\sqrt {b} \sqrt {x} \sqrt {\frac {b x}{a}+1} \left (15 a^2-10 a b x+8 b^2 x^2\right )-15 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{24 b^{7/2} x \sqrt {\frac {b x}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 106, normalized size = 0.85 \begin {gather*} \frac {5 a^3 \log \left (\sqrt {a x^2+b x^3}-\sqrt {b} x^{3/2}\right )}{8 b^{7/2}}-\frac {5 a^3 \log \left (\sqrt {x}\right )}{4 b^{7/2}}+\frac {\left (15 a^2-10 a b x+8 b^2 x^2\right ) \sqrt {a x^2+b x^3}}{24 b^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 180, normalized size = 1.44 \begin {gather*} \left [\frac {15 \, a^{3} \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {b} \sqrt {x}}{x}\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{3} + a x^{2}} \sqrt {x}}{48 \, b^{4} x}, \frac {15 \, a^{3} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-b}}{b x^{\frac {3}{2}}}\right ) + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x^{3} + a x^{2}} \sqrt {x}}{24 \, b^{4} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 64, normalized size = 0.51 \begin {gather*} \frac {1}{24} \, \sqrt {b x + a} {\left (2 \, x {\left (\frac {4 \, x}{b} - \frac {5 \, a}{b^{2}}\right )} + \frac {15 \, a^{2}}{b^{3}}\right )} \sqrt {x} + \frac {5 \, a^{3} \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 103, normalized size = 0.82 \begin {gather*} -\frac {\left (-16 b^{\frac {9}{2}} x^{4}+4 a \,b^{\frac {7}{2}} x^{3}-10 a^{2} b^{\frac {5}{2}} x^{2}-30 a^{3} b^{\frac {3}{2}} x +15 \sqrt {\left (b x +a \right ) x}\, a^{3} b \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )\right ) \sqrt {x}}{48 \sqrt {b \,x^{3}+a \,x^{2}}\, b^{\frac {9}{2}}} \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 {x^{\frac {7}{2}}}{\sqrt {b x^{3} + a x^{2}}}\,{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 {x^{7/2}}{\sqrt {b\,x^3+a\,x^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 {x^{\frac {7}{2}}}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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