Optimal. Leaf size=123 \[ -\frac {x}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {3 d x}{2 \sqrt {c+d x^2} (b c-a d)^2}+\frac {(2 a d+b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 123, 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 = {471, 527, 12, 377, 205} \begin {gather*} -\frac {x}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {3 d x}{2 \sqrt {c+d x^2} (b c-a d)^2}+\frac {(2 a d+b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 471
Rule 527
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\int \frac {c-2 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\int \frac {c (b c+2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 c (b c-a d)^2}\\ &=-\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)^2}\\ &=-\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 (b c-a d)^2}\\ &=-\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.18, size = 133, normalized size = 1.08 \begin {gather*} \frac {x^3 \left (\frac {8 x^2 \left (c+d x^2\right ) (b c-a d) \, _2F_1\left (2,3;\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )}{a+b x^2}+7 c \left (5 c+2 d x^2\right ) \, _2F_1\left (1,2;\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )\right )}{105 c^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.73, size = 135, normalized size = 1.10 \begin {gather*} \frac {(-2 a d-b c) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}}+\frac {-2 a d x-b c x-3 b d x^3}{2 \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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fricas [B] time = 1.73, size = 744, normalized size = 6.05 \begin {gather*} \left [-\frac {{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d + {\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 3 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{4} + {\left (a b^{4} c^{4} - 2 \, a^{2} b^{3} c^{3} d + 2 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )}}, \frac {{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d + {\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (3 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 3 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{4} + {\left (a b^{4} c^{4} - 2 \, a^{2} b^{3} c^{3} d + 2 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.45, size = 299, normalized size = 2.43 \begin {gather*} -\frac {d x}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {d x^{2} + c}} - \frac {{\left (b c \sqrt {d} + 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1453, normalized size = 11.81
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 {x^{2}}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{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 {x^2}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}} \,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 {x^{2}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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