Optimal. Leaf size=143 \[ -\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}}+\frac {b x (a d+2 b c)}{2 a c \sqrt {a+b x^2} (b c-a d)^2}-\frac {d x}{2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {414, 527, 12, 377, 208} \begin {gather*} -\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}}+\frac {b x (a d+2 b c)}{2 a c \sqrt {a+b x^2} (b c-a d)^2}-\frac {d x}{2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 377
Rule 414
Rule 527
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx &=-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-2 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {\int \frac {a d (4 b c-a d)}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 a c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {(d (4 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {(d (4 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 2.67, size = 758, normalized size = 5.30 \begin {gather*} \frac {x \left (\frac {24 d^2 x^4 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {5}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )}{c^2}+\frac {48 d x^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {5}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )}{c}+24 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {5}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+\frac {1050 d^2 x^6 (a d-b c) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c^3 \left (a+b x^2\right )}+\frac {280 d^2 x^4 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}}{c^2}-\frac {2310 d^2 x^4 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}}{c^2}+\frac {2310 d^2 x^4 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c^2}+\frac {2310 d x^4 (a d-b c) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c^2 \left (a+b x^2\right )}+\frac {560 d x^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}}{c}-\frac {5250 d x^2 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}}{c}+70 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}-2625 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}+\frac {5250 d x^2 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c}-\frac {945 x^2 (b c-a d) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c \left (a+b x^2\right )}+2625 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )\right )}{210 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.70, size = 173, normalized size = 1.21 \begin {gather*} \frac {a^2 d^2 x+a b d^2 x^3+2 b^2 c^2 x+2 b^2 c d x^3}{2 a c \sqrt {a+b x^2} \left (c+d x^2\right ) (a d-b c)^2}-\frac {\sqrt {a d-b c} \left (4 b c d-a d^2\right ) \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 )}{2 c^{3/2} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.22, size = 864, normalized size = 6.04 \begin {gather*} \left [-\frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \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 ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 4 \, {\left ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4}\right )} x^{2}\right )}}, \frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} 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 = 1.90, size = 318, normalized size = 2.22 \begin {gather*} \frac {b^{2} x}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b x^{2} + a}} + \frac {{\left (4 \, b^{\frac {3}{2}} c d - a \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{2} + a^{2} \sqrt {b} d^{2}}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1439, normalized size = 10.06
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 {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{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 {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^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 {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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