Optimal. Leaf size=130 \[ \frac {3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 (b c-a d) \left (4 a^2 d^2+(a d+b c)^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^2\right )^2}+\frac {d^3 x}{b^3} \]
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Rubi [A] time = 0.17, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {390, 1157, 385, 205} \begin {gather*} \frac {3 (b c-a d) \left (4 a^2 d^2+(a d+b c)^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}}+\frac {3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^2\right )^2}+\frac {d^3 x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 390
Rule 1157
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac {d^3}{b^3}+\frac {b^3 c^3-a^3 d^3+3 b d (b c-a d) (b c+a d) x^2+3 b^2 d^2 (b c-a d) x^4}{b^3 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac {d^3 x}{b^3}+\frac {\int \frac {b^3 c^3-a^3 d^3+3 b d (b c-a d) (b c+a d) x^2+3 b^2 d^2 (b c-a d) x^4}{\left (a+b x^2\right )^3} \, dx}{b^3}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^2\right )^2}-\frac {\int \frac {-3 (b c-a d) (b c+a d)^2-12 a b d^2 (b c-a d) x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^3}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^2\right )^2}+\frac {3 (b c-a d)^2 (b c+3 a d) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d) \left (4 a^2 d^2+(b c+a d)^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^3}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^2\right )^2}+\frac {3 (b c-a d)^2 (b c+3 a d) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 (b c-a d) \left (4 a^2 d^2+(b c+a d)^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 139, normalized size = 1.07 \begin {gather*} \frac {3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 \left (-5 a^3 d^3+3 a^2 b c d^2+a b^2 c^2 d+b^3 c^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^2\right )^2}+\frac {d^3 x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.64, size = 606, normalized size = 4.66 \begin {gather*} \left [\frac {16 \, a^{3} b^{3} d^{3} x^{5} + 2 \, {\left (3 \, a b^{5} c^{3} + 3 \, a^{2} b^{4} c^{2} d - 15 \, a^{3} b^{3} c d^{2} + 25 \, a^{4} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (b^{5} c^{3} + a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} + a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 9 \, a^{4} b^{2} c d^{2} + 15 \, a^{5} b d^{3}\right )} x}{16 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}, \frac {8 \, a^{3} b^{3} d^{3} x^{5} + {\left (3 \, a b^{5} c^{3} + 3 \, a^{2} b^{4} c^{2} d - 15 \, a^{3} b^{3} c d^{2} + 25 \, a^{4} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (b^{5} c^{3} + a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} + a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 9 \, a^{4} b^{2} c d^{2} + 15 \, a^{5} b d^{3}\right )} x}{8 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 178, normalized size = 1.37 \begin {gather*} \frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} + \frac {3 \, b^{4} c^{3} x^{3} + 3 \, a b^{3} c^{2} d x^{3} - 15 \, a^{2} b^{2} c d^{2} x^{3} + 9 \, a^{3} b d^{3} x^{3} + 5 \, a b^{3} c^{3} x - 3 \, a^{2} b^{2} c^{2} d x - 9 \, a^{3} b c d^{2} x + 7 \, a^{4} d^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 266, normalized size = 2.05 \begin {gather*} \frac {9 a \,d^{3} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {3 c^{2} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a}+\frac {3 b \,c^{3} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {15 c \,d^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {7 a^{2} d^{3} x}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {9 a c \,d^{2} x}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {5 c^{3} x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {3 c^{2} d x}{8 \left (b \,x^{2}+a \right )^{2} b}-\frac {15 a \,d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{3}}+\frac {3 c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {9 c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{2}}+\frac {d^{3} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 185, normalized size = 1.42 \begin {gather*} \frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b^{4} c^{3} + a b^{3} c^{2} d - 5 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} x^{3} + {\left (5 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3}\right )} x}{8 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} + \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 240, normalized size = 1.85 \begin {gather*} \frac {\frac {x\,\left (7\,a^3\,d^3-9\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{8\,a}+\frac {3\,x^3\,\left (3\,a^3\,b\,d^3-5\,a^2\,b^2\,c\,d^2+a\,b^3\,c^2\,d+b^4\,c^3\right )}{8\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {d^3\,x}{b^3}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{\sqrt {a}\,\left (-5\,a^3\,d^3+3\,a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{8\,a^{5/2}\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.89, size = 422, normalized size = 3.25 \begin {gather*} \frac {3 \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log {\left (- \frac {3 a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right )}{15 a^{3} d^{3} - 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{16} - \frac {3 \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log {\left (\frac {3 a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right )}{15 a^{3} d^{3} - 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{16} + \frac {x^{3} \left (9 a^{3} b d^{3} - 15 a^{2} b^{2} c d^{2} + 3 a b^{3} c^{2} d + 3 b^{4} c^{3}\right ) + x \left (7 a^{4} d^{3} - 9 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + 5 a b^{3} c^{3}\right )}{8 a^{4} b^{3} + 16 a^{3} b^{4} x^{2} + 8 a^{2} b^{5} x^{4}} + \frac {d^{3} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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