Optimal. Leaf size=79 \[ \frac {b x}{a \sqrt {a+b x^2} (b c-a d)}-\frac {d \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {382, 377, 208} \[ \frac {b x}{a \sqrt {a+b x^2} (b c-a d)}-\frac {d \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 377
Rule 382
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{b c-a d}\\ &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b c-a d}\\ &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.72, size = 309, normalized size = 3.91 \[ \frac {x \left (2 d x^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+2 c \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-10 d x^2 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}-15 c \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}+10 d x^2 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )+15 c \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )\right )}{5 c^2 \left (a+b x^2\right )^{3/2} \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.07, size = 441, normalized size = 5.58 \[ \left [\frac {4 \, {\left (b^{2} c^{2} - a b c d\right )} \sqrt {b x^{2} + a} x - {\left (a b d x^{2} + a^{2} d\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{2} - a b c d\right )} \sqrt {b x^{2} + a} x + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 107, normalized size = 1.35 \[ -\frac {\sqrt {b} d \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (b c - a d\right )}} + \frac {b x}{{\left (a b c - a^{2} d\right )} \sqrt {b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 618, normalized size = 7.82 \[ -\frac {d \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}+\frac {d \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}-\frac {b x}{2 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}\, a}-\frac {b x}{2 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}\, a}+\frac {d}{2 \sqrt {-c d}\, \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}-\frac {d}{2 \sqrt {-c d}\, \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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