Optimal. Leaf size=152 \[ \frac {4 a^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4}}-\frac {4 a^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{15 b \left (a+b x^2\right )}+\frac {2 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{15 b}+\frac {2}{9} x^3 \left (c \sqrt {a+b x^2}\right )^{3/2} \]
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Rubi [A] time = 0.17, antiderivative size = 191, normalized size of antiderivative = 1.26, 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 = {6720, 279, 321, 229, 227, 196} \[ \frac {4 a^{5/2} c \sqrt [4]{\frac {b x^2}{a}+1} \sqrt {c \sqrt {a+b x^2}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt {a+b x^2}}-\frac {4 a^2 c x \sqrt {c \sqrt {a+b x^2}}}{15 b \sqrt {a+b x^2}}+\frac {2}{9} c x^3 \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}}+\frac {2 a c x \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}}}{15 b} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 229
Rule 279
Rule 321
Rule 6720
Rubi steps
\begin {align*} \int x^2 \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int x^2 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {2}{9} c x^3 \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {\left (a c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {x^2}{\sqrt [4]{a+b x^2}} \, dx}{3 \sqrt [4]{a+b x^2}}\\ &=\frac {2 a c x \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{15 b}+\frac {2}{9} c x^3 \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}-\frac {\left (2 a^2 c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=\frac {2 a c x \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{15 b}+\frac {2}{9} c x^3 \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}-\frac {\left (2 a^2 c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{15 b \sqrt {a+b x^2}}\\ &=-\frac {4 a^2 c x \sqrt {c \sqrt {a+b x^2}}}{15 b \sqrt {a+b x^2}}+\frac {2 a c x \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{15 b}+\frac {2}{9} c x^3 \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {\left (2 a^2 c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{15 b \sqrt {a+b x^2}}\\ &=-\frac {4 a^2 c x \sqrt {c \sqrt {a+b x^2}}}{15 b \sqrt {a+b x^2}}+\frac {2 a c x \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{15 b}+\frac {2}{9} c x^3 \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {4 a^{5/2} c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 68, normalized size = 0.45 \[ \frac {2 x \left (c \sqrt {a+b x^2}\right )^{3/2} \left (-\frac {a \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}+a+b x^2\right )}{9 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{2} + a} \sqrt {\sqrt {b x^{2} + a} c} c x^{2}, x\right ) \]
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 \[ \int \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}} x^{2}\, dx \]
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 \[ \int \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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