Optimal. Leaf size=110 \[ \frac {b^2 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac {\sqrt {c+d x^2}}{3 a c x^3} \]
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Rubi [A] time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {480, 583, 12, 377, 205} \begin {gather*} \frac {b^2 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac {\sqrt {c+d x^2}}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 480
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx &=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {\int \frac {-3 b c-2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c}\\ &=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {\int -\frac {3 b^2 c^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2}\\ &=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \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}\\ &=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 5.12, size = 96, normalized size = 0.87 \begin {gather*} \frac {b^2 \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} \left (-a c+2 a d x^2+3 b c x^2\right )}{3 a^2 c^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 159, normalized size = 1.45 \begin {gather*} \frac {b^2 \sqrt {b c-a d} \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^{5/2} (a d-b c)}+\frac {\sqrt {c+d x^2} \left (-a c+2 a d x^2+3 b c x^2\right )}{3 a^2 c^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.45, size = 414, normalized size = 3.76 \begin {gather*} \left [-\frac {3 \, \sqrt {-a b c + a^{2} d} b^{2} c^{2} x^{3} \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 c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{3}}, \frac {3 \, \sqrt {a b c - a^{2} d} b^{2} c^{2} x^{3} \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 c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.36, size = 195, normalized size = 1.77 \begin {gather*} -\frac {1}{3} \, d^{\frac {5}{2}} {\left (\frac {3 \, b^{2} \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 )}{\sqrt {a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 379, normalized size = 3.45 \begin {gather*} -\frac {b^{2} \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}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {b^{2} \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}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {2 \sqrt {d \,x^{2}+c}\, d}{3 a \,c^{2} x}+\frac {\sqrt {d \,x^{2}+c}\, b}{a^{2} c x}-\frac {\sqrt {d \,x^{2}+c}}{3 a c \,x^{3}} \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 )} \sqrt {d x^{2} + c} x^{4}}\,{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^4\,\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}} \,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^{4} \left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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