Optimal. Leaf size=113 \[ \frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^2}-\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^2}+\frac {b x \sqrt {a+b x^2}}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 = {416, 523, 217, 206, 377, 208} \begin {gather*} \frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^2}-\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^2}+\frac {b x \sqrt {a+b x^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 217
Rule 377
Rule 416
Rule 523
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{c+d x^2} \, dx &=\frac {b x \sqrt {a+b x^2}}{2 d}+\frac {\int \frac {-a (b c-2 a d)-b (2 b c-3 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 d}\\ &=\frac {b x \sqrt {a+b x^2}}{2 d}-\frac {(b (2 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{d^2}\\ &=\frac {b x \sqrt {a+b x^2}}{2 d}-\frac {(b (2 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 d^2}+\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{d^2}\\ &=\frac {b x \sqrt {a+b x^2}}{2 d}-\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^2}+\frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 110, normalized size = 0.97 \begin {gather*} \frac {\sqrt {b} (3 a d-2 b c) \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )+\frac {2 (a d-b c)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {a d-b c}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}+b d x \sqrt {a+b x^2}}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 148, normalized size = 1.31 \begin {gather*} \frac {\left (2 b^{3/2} c-3 a \sqrt {b} d\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{2 d^2}+\frac {(b c-a d) \sqrt {a d-b c} \tan ^{-1}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} c+\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}\right )}{\sqrt {c} d^2}+\frac {b x \sqrt {a+b x^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 721, normalized size = 6.38 \begin {gather*} \left [\frac {2 \, \sqrt {b x^{2} + a} b d x - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac {2 \, \sqrt {b x^{2} + a} b d x + 2 \, {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac {2 \, \sqrt {b x^{2} + a} b d x - 2 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, d^{2}}, \frac {\sqrt {b x^{2} + a} b d x + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right )}{2 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1845, normalized size = 16.33
result too large to display
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*} \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{d x^{2} + c}\,{d x} \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 \frac {{\left (b\,x^2+a\right )}^{3/2}}{d\,x^2+c} \,d x \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 \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{c + d x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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