Optimal. Leaf size=163 \[ \frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.12, antiderivative size = 163, 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 {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (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}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx &=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-2 b d x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\int \frac {8 b^2 c^2-8 a b c d+3 a^2 d^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2}\\ &=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2}\\ &=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^2}\\ &=-\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 192, normalized size = 1.18 \begin {gather*} \frac {x \left (\frac {\left (c+d x^2\right )^2 \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}}-c d \left (a^2 (-d) \left (5 c+3 d x^2\right )+a b \left (8 c^2+c d x^2-3 d^2 x^4\right )+2 b^2 c x^2 \left (4 c+3 d x^2\right )\right )\right )}{8 c^3 \sqrt {a+b x^2} \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.91, size = 180, normalized size = 1.10 \begin {gather*} \frac {\sqrt {a d-b c} \left (3 a^2 d^2-8 a b c d+8 b^2 c^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 )}{8 c^{5/2} (b c-a d)^3}+\frac {\sqrt {a+b x^2} \left (5 a c d^2 x+3 a d^3 x^3-8 b c^2 d x-6 b c d^2 x^3\right )}{8 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.97, size = 864, normalized size = 5.30 \begin {gather*} \left [\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c 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 (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} x^{2}\right )}}, -\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c 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 (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} 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 = 3.51, size = 538, normalized size = 3.30 \begin {gather*} -\frac {1}{8} \, b^{\frac {5}{2}} {\left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} 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 )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{2} c^{2} d - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b c d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} d^{3} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{3} c^{3} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{2} c^{2} d + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} d^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{2} c^{2} d - 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} d^{3} + 6 \, a^{4} b c d^{2} - 3 \, a^{5} d^{3}\right )}}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} 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 )}^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1815, normalized size = 11.13
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}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3}}\,{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}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3} \,d x \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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