Optimal. Leaf size=161 \[ \frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}-\frac {x^5}{b \sqrt {a x+b x^3}} \]
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Rubi [A] time = 0.19, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2022, 2024, 2011, 329, 220} \[ \frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}-\frac {x^5}{b \sqrt {a x+b x^3}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2022
Rule 2024
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx &=-\frac {x^5}{b \sqrt {a x+b x^3}}+\frac {9 \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx}{2 b}\\ &=-\frac {x^5}{b \sqrt {a x+b x^3}}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}-\frac {(45 a) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{14 b^2}\\ &=-\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{14 b^3}\\ &=-\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{14 b^3 \sqrt {a x+b x^3}}\\ &=-\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{7 b^3 \sqrt {a x+b x^3}}\\ &=-\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 80, normalized size = 0.50 \[ \frac {x \left (15 a^2 \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )-15 a^2-6 a b x^2+2 b^2 x^4\right )}{7 b^3 \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x} x^{5}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{7}}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 172, normalized size = 1.07 \[ -\frac {a^{2} x}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}\, b^{3}}+\frac {2 \sqrt {b \,x^{3}+a x}\, x^{2}}{7 b^{2}}+\frac {15 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{2} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14 \sqrt {b \,x^{3}+a x}\, b^{4}}-\frac {8 \sqrt {b \,x^{3}+a x}\, a}{7 b^{3}} \]
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 \[ \int \frac {x^{7}}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^7}{{\left (b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{7}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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