Optimal. Leaf size=293 \[ -\frac {2 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4} \]
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Rubi [A] time = 0.31, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2021, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}+\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {b x^2+c x^4} \, dx &=\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}+\frac {1}{9} (2 b) \int \frac {x^{7/2}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}-\frac {\left (2 b^2\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 c}\\ &=\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}-\frac {\left (2 b^2 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 c \sqrt {b x^2+c x^4}}\\ &=\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}-\frac {\left (4 b^2 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 c \sqrt {b x^2+c x^4}}\\ &=\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}-\frac {\left (4 b^{5/2} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 c^{3/2} \sqrt {b x^2+c x^4}}+\frac {\left (4 b^{5/2} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 c^{3/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4 b \sqrt {x} \sqrt {b x^2+c x^4}}{45 c}+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}+\frac {4 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {2 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 86, normalized size = 0.29 \[ \frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right ) \sqrt {\frac {c x^2}{b}+1}-b \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )\right )}{9 c \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{4} + b x^{2}} x^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sqrt {c x^{4} + b x^{2}} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 226, normalized size = 0.77 \[ -\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (-5 c^{3} x^{6}-7 b \,c^{2} x^{4}-2 b^{2} c \,x^{2}+6 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right )}{45 \left (c \,x^{2}+b \right ) c^{2} x^{\frac {3}{2}}} \]
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 \[ \int \sqrt {c x^{4} + b x^{2}} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^{3/2}\,\sqrt {c\,x^4+b\,x^2} \,d x \]
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 \[ \int x^{\frac {3}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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