Optimal. Leaf size=350 \[ \frac {4 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{5 x} \]
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Rubi [A] time = 0.43, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2018, 2020, 2025, 2032, 329, 305, 220, 1196} \[ \frac {4 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{5 x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2020
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+(2 a) \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {1}{5} \left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 b}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (4 a^3 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 b \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (8 a^3 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 b \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (8 a^{5/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (8 a^{5/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {8 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}-\frac {8 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {4 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 62, normalized size = 0.18 \[ -\frac {2 b \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {a x^{2/3}}{b}\right )}{3 x^{5/3} \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 339, normalized size = 0.97 \[ -\frac {2 \left (-12 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+12 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{3} x^{\frac {10}{3}}+12 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}}+11 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b \,x^{\frac {8}{3}}+16 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{2} x^{2}+5 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{3} x^{\frac {4}{3}}\right )}{15 \left (a \,x^{\frac {2}{3}}+b \right ) b \,x^{3}} \]
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 \[ \int \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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