Optimal. Leaf size=176 \[ \frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)} \]
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Rubi [A] time = 0.22, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {472, 583, 12, 377, 205} \[ \frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 472
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}+\frac {\int \frac {b c-4 a d-4 b d x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{c (b c-a d)}\\ &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}-\frac {\int \frac {(3 b c-4 a d) (b c+2 a d)+2 b d (b c-4 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c^2 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {\int \frac {3 b^3 c^3}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a^2 c^3 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) 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^2 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 5.25, size = 124, normalized size = 0.70 \[ \frac {b^3 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^2} \left (\frac {x^2 (5 a d+3 b c)}{a^2}+\frac {3 d^3 x^4}{\left (c+d x^2\right ) (a d-b c)}-\frac {c}{a}\right )}{3 c^3 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.58, size = 706, normalized size = 4.01 \[ \left [\frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}, \frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.62, size = 275, normalized size = 1.56 \[ \frac {b^{3} \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {d^{3} x}{{\left (b c^{4} - a c^{3} d\right )} \sqrt {d x^{2} + c}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 5 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 762, normalized size = 4.33 \[ \frac {b^{3} \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a^{2}}-\frac {b^{3} \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {b^{3}}{2 \sqrt {-a b}\, \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2}}-\frac {b^{3}}{2 \sqrt {-a b}\, \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2}}+\frac {b^{2} d x}{2 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2} c}+\frac {b^{2} d x}{2 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2} c}+\frac {8 d^{2} x}{3 \sqrt {d \,x^{2}+c}\, a \,c^{3}}+\frac {2 b d x}{\sqrt {d \,x^{2}+c}\, a^{2} c^{2}}+\frac {4 d}{3 \sqrt {d \,x^{2}+c}\, a \,c^{2} x}+\frac {b}{\sqrt {d \,x^{2}+c}\, a^{2} c x}-\frac {1}{3 \sqrt {d \,x^{2}+c}\, a c \,x^{3}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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