Optimal. Leaf size=152 \[ \frac {2 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{15 b}+\frac {2}{9} x^3 \left (c \sqrt {a+b x^2}\right )^{3/2}-\frac {4 a^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{15 b \left (a+b x^2\right )}+\frac {4 a^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1973, 285, 327,
233, 202} \begin {gather*} \frac {4 a^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 285
Rule 327
Rule 1973
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.58, size = 68, normalized size = 0.45 \begin {gather*} \frac {2 x \left (c \sqrt {a+b x^2}\right )^{3/2} \left (a+b x^2-\frac {a \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{3/4}}\right )}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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