Optimal. Leaf size=178 \[ -\frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}-\frac {\sqrt {c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac {d (7 b c-4 a d)}{3 c^2 x \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c x \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.24, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {472, 579, 583, 12, 377, 205} \begin {gather*} -\frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}-\frac {\sqrt {c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac {d (7 b c-4 a d)}{3 c^2 x \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c x \left (c+d x^2\right )^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 472
Rule 579
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {3 b c-4 a d-4 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)}\\ &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}+\frac {\int \frac {(b c-4 a d) (3 b c-2 a d)-2 b d (7 b c-4 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c^2 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {\int \frac {3 b^3 c^3}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c^3 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x \left (c+d x^2\right )^{3/2}}-\frac {d (7 b c-4 a d)}{3 c^2 (b c-a d)^2 x \sqrt {c+d x^2}}-\frac {(b c-4 a d) (3 b c-2 a d) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x}-\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 5.27, size = 143, normalized size = 0.80 \begin {gather*} \frac {\sqrt {c+d x^2} \left (\frac {d^2 x^2 (8 b c-5 a d)}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {c d^2 x^2}{\left (c+d x^2\right )^2 (b c-a d)}-\frac {3}{a}\right )}{3 c^3 x}-\frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 241, normalized size = 1.35 \begin {gather*} \frac {b^3 \tan ^{-1}\left (\frac {b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}-\frac {b x \sqrt {c+d x^2}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{a^{3/2} (b c-a d)^{5/2}}+\frac {-3 a^2 c^2 d^2-12 a^2 c d^3 x^2-8 a^2 d^4 x^4+6 a b c^3 d+21 a b c^2 d^2 x^2+14 a b c d^3 x^4-3 b^2 c^4-6 b^2 c^3 d x^2-3 b^2 c^2 d^2 x^4}{3 a c^3 x \left (c+d x^2\right )^{3/2} (a d-b c)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.17, size = 934, normalized size = 5.25 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} x\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 \, a b^{3} c^{5} - 9 \, a^{2} b^{2} c^{4} d + 9 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} - 17 \, a^{2} b^{2} c^{2} d^{3} + 22 \, a^{3} b c d^{4} - 8 \, a^{4} d^{5}\right )} x^{4} + 3 \, {\left (2 \, a b^{3} c^{4} d - 9 \, a^{2} b^{2} c^{3} d^{2} + 11 \, a^{3} b c^{2} d^{3} - 4 \, a^{4} c d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )} x^{5} + 2 \, {\left (a^{2} b^{3} c^{7} d - 3 \, a^{3} b^{2} c^{6} d^{2} + 3 \, a^{4} b c^{5} d^{3} - a^{5} c^{4} d^{4}\right )} x^{3} + {\left (a^{2} b^{3} c^{8} - 3 \, a^{3} b^{2} c^{7} d + 3 \, a^{4} b c^{6} d^{2} - a^{5} c^{5} d^{3}\right )} x\right )}}, -\frac {3 \, {\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} x\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 \, a b^{3} c^{5} - 9 \, a^{2} b^{2} c^{4} d + 9 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} - 17 \, a^{2} b^{2} c^{2} d^{3} + 22 \, a^{3} b c d^{4} - 8 \, a^{4} d^{5}\right )} x^{4} + 3 \, {\left (2 \, a b^{3} c^{4} d - 9 \, a^{2} b^{2} c^{3} d^{2} + 11 \, a^{3} b c^{2} d^{3} - 4 \, a^{4} c d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )} x^{5} + 2 \, {\left (a^{2} b^{3} c^{7} d - 3 \, a^{3} b^{2} c^{6} d^{2} + 3 \, a^{4} b c^{5} d^{3} - a^{5} c^{4} d^{4}\right )} x^{3} + {\left (a^{2} b^{3} c^{8} - 3 \, a^{3} b^{2} c^{7} d + 3 \, a^{4} b c^{6} d^{2} - a^{5} c^{5} d^{3}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.80, size = 366, normalized size = 2.06 \begin {gather*} \frac {b^{3} \sqrt {d} \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 )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\frac {{\left (8 \, b^{3} c^{5} d^{4} - 21 \, a b^{2} c^{4} d^{5} + 18 \, a^{2} b c^{3} d^{6} - 5 \, a^{3} c^{2} d^{7}\right )} x^{2}}{b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}} + \frac {3 \, {\left (3 \, b^{3} c^{6} d^{3} - 8 \, a b^{2} c^{5} d^{4} + 7 \, a^{2} b c^{4} d^{5} - 2 \, a^{3} c^{3} d^{6}\right )}}{b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {2 \, \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1192, normalized size = 6.70 \begin {gather*} \frac {b^{3} \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {b^{3} \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {b^{3}}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}-\frac {b^{3}}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}+\frac {b^{2} d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a c}+\frac {b^{2} d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a c}-\frac {b^{2}}{6 \sqrt {-a b}\, \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a}+\frac {b^{2}}{6 \sqrt {-a b}\, \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a}-\frac {b d x}{6 \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a c}-\frac {b d x}{6 \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a c}-\frac {b d x}{3 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a \,c^{2}}-\frac {b d x}{3 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a \,c^{2}}-\frac {4 d x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a \,c^{2}}-\frac {8 d x}{3 \sqrt {d \,x^{2}+c}\, a \,c^{3}}-\frac {1}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} a c x} \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 )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{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}{x^2\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,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}{x^{2} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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