Optimal. Leaf size=66 \[ \frac {2 \left (a+b x^2\right )^2 \left (c \sqrt {a+b x^2}\right )^{3/2}}{11 b^2}-\frac {2 a \left (a+b x^2\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{7 b^2} \]
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Rubi [A] time = 0.14, antiderivative size = 74, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6720, 266, 43} \[ \frac {2 c \left (a+b x^2\right )^{5/2} \sqrt {c \sqrt {a+b x^2}}}{11 b^2}-\frac {2 a c \left (a+b x^2\right )^{3/2} \sqrt {c \sqrt {a+b x^2}}}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6720
Rubi steps
\begin {align*} \int x^3 \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int x^3 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int x (a+b x)^{3/4} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^{3/4}}{b}+\frac {(a+b x)^{7/4}}{b}\right ) \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 a c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b^2}+\frac {2 c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{5/2}}{11 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.62 \[ \frac {2 \left (a+b x^2\right ) \left (7 b x^2-4 a\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{77 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 51, normalized size = 0.77 \[ \frac {2 \, {\left (7 \, b^{2} c x^{4} + 3 \, a b c x^{2} - 4 \, a^{2} c\right )} \sqrt {b x^{2} + a} \sqrt {\sqrt {b x^{2} + a} c}}{77 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 81, normalized size = 1.23 \[ \frac {2 \, {\left (\frac {11 \, {\left (3 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} - 7 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a\right )} a}{b} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} - 66 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a + 77 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{2}}{b}\right )} c^{\frac {3}{2}}}{231 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.55 \[ -\frac {2 \left (b \,x^{2}+a \right ) \left (-7 b \,x^{2}+4 a \right ) \left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}}}{77 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 43, normalized size = 0.65 \[ -\frac {2 \, {\left (11 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {7}{2}} a c^{2} - 7 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {11}{2}}\right )}}{77 \, b^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.89, size = 67, normalized size = 1.02 \[ \sqrt {c\,\sqrt {b\,x^2+a}}\,\left (\frac {2\,c\,x^4\,\sqrt {b\,x^2+a}}{11}-\frac {8\,a^2\,c\,\sqrt {b\,x^2+a}}{77\,b^2}+\frac {6\,a\,c\,x^2\,\sqrt {b\,x^2+a}}{77\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.30, size = 87, normalized size = 1.32 \[ \begin {cases} - \frac {8 a^{2} c^{\frac {3}{2}} \left (a + b x^{2}\right )^{\frac {3}{4}}}{77 b^{2}} + \frac {6 a c^{\frac {3}{2}} x^{2} \left (a + b x^{2}\right )^{\frac {3}{4}}}{77 b} + \frac {2 c^{\frac {3}{2}} x^{4} \left (a + b x^{2}\right )^{\frac {3}{4}}}{11} & \text {for}\: b \neq 0 \\\frac {x^{4} \left (\sqrt {a} c\right )^{\frac {3}{2}}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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