Optimal. Leaf size=134 \[ \frac {a}{2 b \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {a d+2 b c}{2 b \sqrt {c+d x^2} (b c-a d)^2}-\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 78, 51, 63, 208} \begin {gather*} \frac {a}{2 b \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {a d+2 b c}{2 b \sqrt {c+d x^2} (b c-a d)^2}-\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 b (b c-a d)}\\ &=\frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^2}\\ &=\frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 d (b c-a d)^2}\\ &=\frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 91, normalized size = 0.68 \begin {gather*} \frac {\left (a+b x^2\right ) (a d+2 b c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \left (d x^2+c\right )}{b c-a d}\right )+a (b c-a d)}{2 b \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 123, normalized size = 0.92 \begin {gather*} \frac {3 a c+a d x^2+2 b c x^2}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)^2}+\frac {(-a d-2 b c) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{2 \sqrt {b} (a d-b c)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.18, size = 732, normalized size = 5.46 \begin {gather*} \left [\frac {{\left ({\left (2 \, b^{2} c d + a b d^{2}\right )} x^{4} + 2 \, a b c^{2} + a^{2} c d + {\left (2 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + {\left (2 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{4} c^{4} - 3 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} - a^{4} b c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{4} + {\left (b^{5} c^{4} - 2 \, a b^{4} c^{3} d + 2 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{2}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c d + a b d^{2}\right )} x^{4} + 2 \, a b c^{2} + a^{2} c d + {\left (2 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + {\left (2 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{4} c^{4} - 3 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} - a^{4} b c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{4} + {\left (b^{5} c^{4} - 2 \, a b^{4} c^{3} d + 2 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 181, normalized size = 1.35 \begin {gather*} \frac {\frac {{\left (2 \, b c d + a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (d x^{2} + c\right )} b c d - 2 \, b c^{2} d + {\left (d x^{2} + c\right )} a d^{2} + 2 \, a c d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{2} + c} b c + \sqrt {d x^{2} + c} a d\right )}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1456, normalized size = 10.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 142, normalized size = 1.06 \begin {gather*} \frac {\frac {c}{a\,d-b\,c}+\frac {\left (d\,x^2+c\right )\,\left (a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^2}}{b\,{\left (d\,x^2+c\right )}^{3/2}+\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )\,\left (a\,d+2\,b\,c\right )}{2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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