Optimal. Leaf size=122 \[ \frac {b x (2 b c-5 a d)}{3 a^2 \sqrt {a+b x^2} (b c-a d)^2}+\frac {d^2 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{5/2}}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.10, antiderivative size = 122, 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 {b x (2 b c-5 a d)}{3 a^2 \sqrt {a+b x^2} (b c-a d)^2}+\frac {d^2 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{5/2}}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} (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}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )} \, dx &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-2 b c+3 a d-2 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx}{3 a (b c-a d)}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {b (2 b c-5 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {\int \frac {3 a^2 d^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{3 a^2 (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {b (2 b c-5 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {d^2 \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{(b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {b (2 b c-5 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {d^2 \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)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {b (2 b c-5 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 2.75, size = 775, normalized size = 6.35 \begin {gather*} \frac {x \left (12 c^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 {7}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+12 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 {7}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+24 c 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 {7}{2};1,\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+48 c^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-105 c^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}-315 c^2 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}+315 c^2 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )+36 d^2 x^4 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-56 d^2 x^4 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}-168 d^2 x^4 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}+168 d^2 x^4 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )+84 c d x^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-140 c d x^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}-420 c d x^2 \sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}+420 c d x^2 \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )\right )}{63 c^3 \left (a+b x^2\right )^{5/2} \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 = 0.41, size = 180, normalized size = 1.48 \begin {gather*} \frac {-6 a^2 b d x+3 a b^2 c x-5 a b^2 d x^3+2 b^3 c x^3}{3 a^2 \left (a+b x^2\right )^{3/2} (a d-b c)^2}+\frac {d^2 \sqrt {a d-b c} \tan ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}-\frac {d x \sqrt {a+b x^2}}{\sqrt {c} \sqrt {a d-b c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a d-b c}}\right )}{\sqrt {c} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.74, size = 764, normalized size = 6.26 \begin {gather*} \left [\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{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^{4} c^{3} - 7 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 2 \, a^{3} b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{4} b^{3} c^{4} - 3 \, a^{5} b^{2} c^{3} d + 3 \, a^{6} b c^{2} d^{2} - a^{7} c d^{3} + {\left (a^{2} b^{5} c^{4} - 3 \, a^{3} b^{4} c^{3} d + 3 \, a^{4} b^{3} c^{2} d^{2} - a^{5} b^{2} c d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{4} - 3 \, a^{4} b^{3} c^{3} d + 3 \, a^{5} b^{2} c^{2} d^{2} - a^{6} b c d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{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^{4} c^{3} - 7 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 2 \, a^{3} b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{4} b^{3} c^{4} - 3 \, a^{5} b^{2} c^{3} d + 3 \, a^{6} b c^{2} d^{2} - a^{7} c d^{3} + {\left (a^{2} b^{5} c^{4} - 3 \, a^{3} b^{4} c^{3} d + 3 \, a^{4} b^{3} c^{2} d^{2} - a^{5} b^{2} c d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{4} - 3 \, a^{4} b^{3} c^{3} d + 3 \, a^{5} b^{2} c^{2} d^{2} - a^{6} b c d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 320, normalized size = 2.62 \begin {gather*} -\frac {\sqrt {b} d^{2} \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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {{\left (\frac {{\left (2 \, b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 12 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{2}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}} + \frac {3 \, {\left (a b^{5} c^{3} - 4 \, a^{2} b^{4} c^{2} d + 5 \, a^{3} b^{3} c d^{2} - 2 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1070, normalized size = 8.77 \begin {gather*} -\frac {d^{2} \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 )^{2} \sqrt {\frac {a d -b c}{d}}}+\frac {d^{2} \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 )^{2} \sqrt {\frac {a d -b c}{d}}}-\frac {b d x}{2 \left (a d -b c \right )^{2} \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 d x}{2 \left (a d -b c \right )^{2} \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}}{2 \sqrt {-c d}\, \left (a d -b c \right )^{2} \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}}{2 \sqrt {-c d}\, \left (a d -b c \right )^{2} \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 {b x}{6 \left (a d -b c \right ) \left (\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}\right )^{\frac {3}{2}} a}-\frac {b x}{6 \left (a d -b c \right ) \left (\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}\right )^{\frac {3}{2}} a}+\frac {d}{6 \sqrt {-c d}\, \left (a d -b c \right ) \left (\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}\right )^{\frac {3}{2}}}-\frac {d}{6 \sqrt {-c d}\, \left (a d -b c \right ) \left (\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}\right )^{\frac {3}{2}}}-\frac {b x}{3 \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^{2}}-\frac {b x}{3 \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^{2}} \end {gather*}
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 {5}{2}} {\left (d x^{2} + c\right )}}\,{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 )}^{5/2}\,\left (d\,x^2+c\right )} \,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 {5}{2}} \left (c + d x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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