Optimal. Leaf size=709 \[ -\frac {8 \sqrt {2} 3^{3/4} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {24 a^2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b^{5/3} \left (c x^2\right )^{5/2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}+\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.49, antiderivative size = 709, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {368, 279, 321, 303, 218, 1877} \[ -\frac {24 a^2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b^{5/3} \left (c x^2\right )^{5/2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}-\frac {8 \sqrt {2} 3^{3/4} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}+\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 303
Rule 321
Rule 368
Rule 1877
Rubi steps
\begin {align*} \int x^4 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x^5 \operatorname {Subst}\left (\int x^4 \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{\left (c x^2\right )^{5/2}}\\ &=\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {\left (3 a x^5\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{13 \left (c x^2\right )^{5/2}}\\ &=\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}-\frac {\left (12 a^2 x^5\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{91 b \left (c x^2\right )^{5/2}}\\ &=\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}-\frac {\left (12 a^2 x^5\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{91 b^{4/3} \left (c x^2\right )^{5/2}}-\frac {\left (12 \sqrt {2 \left (2-\sqrt {3}\right )} a^{7/3} x^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{91 b^{4/3} \left (c x^2\right )^{5/2}}\\ &=\frac {2}{13} x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {6 a c x^7 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b \left (c x^2\right )^{5/2}}-\frac {24 a^2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}}}{91 b^{5/3} \left (c x^2\right )^{5/2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {8 \sqrt {2} 3^{3/4} a^{7/3} x^5 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \left (c x^2\right )^{5/2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 109, normalized size = 0.15 \[ \frac {2 x^5 \sqrt {a+b \left (c x^2\right )^{3/2}} \left (a \left (\frac {a+b \left (c x^2\right )^{3/2}}{a}\right )^{3/2}-a \, _2F_1\left (-\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b \left (c x^2\right )^{3/2}}{a}\right )\right )}{13 b \left (c x^2\right )^{3/2} \sqrt {\frac {a+b \left (c x^2\right )^{3/2}}{a}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sqrt {c x^{2}} b c x^{2} + a} x^{4}, x\right ) \]
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 \[ \int \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}\, x^{4}\, dx \]
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 \[ \int \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} x^{4}\,{d x} \]
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 \[ \int x^4\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int x^{4} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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