Optimal. Leaf size=163 \[ -\frac {4 b^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 a^{5/4} \sqrt {a x+b x^3}}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac {12 b \sqrt {a x+b x^3}}{77 x^4} \]
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Rubi [A] time = 0.18, antiderivative size = 163, 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 = {2020, 2025, 2011, 329, 220} \[ -\frac {4 b^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 a^{5/4} \sqrt {a x+b x^3}}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3\right )^{3/2}}{x^8} \, dx &=-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}+\frac {1}{11} (6 b) \int \frac {\sqrt {a x+b x^3}}{x^5} \, dx\\ &=-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}+\frac {1}{77} \left (12 b^2\right ) \int \frac {1}{x^2 \sqrt {a x+b x^3}} \, dx\\ &=-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac {\left (4 b^3\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{77 a}\\ &=-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac {\left (4 b^3 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{77 a \sqrt {a x+b x^3}}\\ &=-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac {\left (8 b^3 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{77 a \sqrt {a x+b x^3}}\\ &=-\frac {12 b \sqrt {a x+b x^3}}{77 x^4}-\frac {8 b^2 \sqrt {a x+b x^3}}{77 a x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac {4 b^{11/4} \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 )}{77 a^{5/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.33 \[ -\frac {2 a \sqrt {x \left (a+b x^2\right )} \, _2F_1\left (-\frac {11}{4},-\frac {3}{2};-\frac {7}{4};-\frac {b x^2}{a}\right )}{11 x^6 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x} {\left (b x^{2} + a\right )}}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 169, normalized size = 1.04 \[ -\frac {4 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, b^{2} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 \sqrt {b \,x^{3}+a x}\, a}-\frac {8 \sqrt {b \,x^{3}+a x}\, b^{2}}{77 a \,x^{2}}-\frac {26 \sqrt {b \,x^{3}+a x}\, b}{77 x^{4}}-\frac {2 \sqrt {b \,x^{3}+a x}\, a}{11 x^{6}} \]
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 \[ \int \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x\right )}^{3/2}}{x^8} \,d x \]
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 \[ \int \frac {\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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