Optimal. Leaf size=127 \[ -\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}}+\frac {x (b c-a d) (a d+7 b c)}{8 c d^3 \left (c+d x^2\right )}+\frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {b^2 x}{d^3} \]
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Rubi [A] time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {463, 455, 388, 205} \[ -\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}}+\frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {x (b c-a d) (a d+7 b c)}{8 c d^3 \left (c+d x^2\right )}+\frac {b^2 x}{d^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 455
Rule 463
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac {(b c-a d)^2 x^3}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^2 \left (-4 a^2 d^2+3 (b c-a d)^2-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac {(b c-a d)^2 x^3}{4 c d^2 \left (c+d x^2\right )^2}+\frac {(b c-a d) (7 b c+a d) x}{8 c d^3 \left (c+d x^2\right )}+\frac {\int \frac {-d (b c-a d) (7 b c+a d)+8 b^2 c d^2 x^2}{c+d x^2} \, dx}{8 c d^4}\\ &=\frac {b^2 x}{d^3}+\frac {(b c-a d)^2 x^3}{4 c d^2 \left (c+d x^2\right )^2}+\frac {(b c-a d) (7 b c+a d) x}{8 c d^3 \left (c+d x^2\right )}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 c d^3}\\ &=\frac {b^2 x}{d^3}+\frac {(b c-a d)^2 x^3}{4 c d^2 \left (c+d x^2\right )^2}+\frac {(b c-a d) (7 b c+a d) x}{8 c d^3 \left (c+d x^2\right )}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 130, normalized size = 1.02 \[ \frac {x \left (a^2 d^2 \left (d x^2-c\right )-2 a b c d \left (3 c+5 d x^2\right )+b^2 c \left (15 c^2+25 c d x^2+8 d^2 x^4\right )\right )}{8 c d^3 \left (c+d x^2\right )^2}-\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 475, normalized size = 3.74 \[ \left [\frac {16 \, b^{2} c^{2} d^{3} x^{5} + 2 \, {\left (25 \, b^{2} c^{3} d^{2} - 10 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (15 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} x}{16 \, {\left (c^{2} d^{6} x^{4} + 2 \, c^{3} d^{5} x^{2} + c^{4} d^{4}\right )}}, \frac {8 \, b^{2} c^{2} d^{3} x^{5} + {\left (25 \, b^{2} c^{3} d^{2} - 10 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} - {\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (15 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} x}{8 \, {\left (c^{2} d^{6} x^{4} + 2 \, c^{3} d^{5} x^{2} + c^{4} d^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 133, normalized size = 1.05 \[ \frac {b^{2} x}{d^{3}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c d^{3}} + \frac {9 \, b^{2} c^{2} d x^{3} - 10 \, a b c d^{2} x^{3} + a^{2} d^{3} x^{3} + 7 \, b^{2} c^{3} x - 6 \, a b c^{2} d x - a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 196, normalized size = 1.54 \[ \frac {a^{2} x^{3}}{8 \left (d \,x^{2}+c \right )^{2} c}-\frac {5 a b \,x^{3}}{4 \left (d \,x^{2}+c \right )^{2} d}+\frac {9 b^{2} c \,x^{3}}{8 \left (d \,x^{2}+c \right )^{2} d^{2}}-\frac {a^{2} x}{8 \left (d \,x^{2}+c \right )^{2} d}-\frac {3 a b c x}{4 \left (d \,x^{2}+c \right )^{2} d^{2}}+\frac {7 b^{2} c^{2} x}{8 \left (d \,x^{2}+c \right )^{2} d^{3}}+\frac {a^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c d}+\frac {3 a b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \sqrt {c d}\, d^{2}}-\frac {15 b^{2} c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, d^{3}}+\frac {b^{2} x}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 143, normalized size = 1.13 \[ \frac {{\left (9 \, b^{2} c^{2} d - 10 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (7 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2}\right )} x}{8 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} + \frac {b^{2} x}{d^{3}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 135, normalized size = 1.06 \[ \frac {b^2\,x}{d^3}-\frac {x\,\left (\frac {a^2\,d^2}{8}+\frac {3\,a\,b\,c\,d}{4}-\frac {7\,b^2\,c^2}{8}\right )-\frac {x^3\,\left (a^2\,d^3-10\,a\,b\,c\,d^2+9\,b^2\,c^2\,d\right )}{8\,c}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (a^2\,d^2+6\,a\,b\,c\,d-15\,b^2\,c^2\right )}{8\,c^{3/2}\,d^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.50, size = 223, normalized size = 1.76 \[ \frac {b^{2} x}{d^{3}} - \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log {\left (- c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log {\left (c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} + x \right )}}{16} + \frac {x^{3} \left (a^{2} d^{3} - 10 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x \left (- a^{2} c d^{2} - 6 a b c^{2} d + 7 b^{2} c^{3}\right )}{8 c^{3} d^{3} + 16 c^{2} d^{4} x^{2} + 8 c d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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