Optimal. Leaf size=118 \[ \frac {a^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}+\frac {x \left (a+b x^2\right )^{3/2} (6 b c-a d)}{24 b}+\frac {a x \sqrt {a+b x^2} (6 b c-a d)}{16 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b} \]
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Rubi [A] time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {388, 195, 217, 206} \[ \frac {a^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}+\frac {x \left (a+b x^2\right )^{3/2} (6 b c-a d)}{24 b}+\frac {a x \sqrt {a+b x^2} (6 b c-a d)}{16 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \, dx &=\frac {d x \left (a+b x^2\right )^{5/2}}{6 b}-\frac {(-6 b c+a d) \int \left (a+b x^2\right )^{3/2} \, dx}{6 b}\\ &=\frac {(6 b c-a d) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {(a (6 b c-a d)) \int \sqrt {a+b x^2} \, dx}{8 b}\\ &=\frac {a (6 b c-a d) x \sqrt {a+b x^2}}{16 b}+\frac {(6 b c-a d) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b}\\ &=\frac {a (6 b c-a d) x \sqrt {a+b x^2}}{16 b}+\frac {(6 b c-a d) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b}\\ &=\frac {a (6 b c-a d) x \sqrt {a+b x^2}}{16 b}+\frac {(6 b c-a d) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {d x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 109, normalized size = 0.92 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (3 a^2 d+2 a b \left (15 c+7 d x^2\right )+4 b^2 x^2 \left (3 c+2 d x^2\right )\right )-\frac {3 a^{3/2} (a d-6 b c) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{48 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 210, normalized size = 1.78 \[ \left [-\frac {3 \, {\left (6 \, a^{2} b c - a^{3} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, b^{3} d x^{5} + 2 \, {\left (6 \, b^{3} c + 7 \, a b^{2} d\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c + a^{2} b d\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{2}}, -\frac {3 \, {\left (6 \, a^{2} b c - a^{3} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{3} d x^{5} + 2 \, {\left (6 \, b^{3} c + 7 \, a b^{2} d\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c + a^{2} b d\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 103, normalized size = 0.87 \[ \frac {1}{48} \, {\left (2 \, {\left (4 \, b d x^{2} + \frac {6 \, b^{5} c + 7 \, a b^{4} d}{b^{4}}\right )} x^{2} + \frac {3 \, {\left (10 \, a b^{4} c + a^{2} b^{3} d\right )}}{b^{4}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (6 \, a^{2} b c - a^{3} d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 131, normalized size = 1.11 \[ -\frac {a^{3} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {3}{2}}}+\frac {3 a^{2} c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}-\frac {\sqrt {b \,x^{2}+a}\, a^{2} d x}{16 b}+\frac {3 \sqrt {b \,x^{2}+a}\, a c x}{8}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} a d x}{24 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} c x}{4}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} d x}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 116, normalized size = 0.98 \[ \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c x + \frac {3}{8} \, \sqrt {b x^{2} + a} a c x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a d x}{24 \, b} - \frac {\sqrt {b x^{2} + a} a^{2} d x}{16 \, b} + \frac {3 \, a^{2} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - \frac {a^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} \]
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 \[ \int {\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.71, size = 253, normalized size = 2.14 \[ \frac {a^{\frac {5}{2}} d x}{16 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {3}{2}} c x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} c x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} d x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 \sqrt {a} b c x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {11 \sqrt {a} b d x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{3} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + \frac {3 a^{2} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {b^{2} c x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{2} d x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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