Optimal. Leaf size=160 \[ \frac {x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}}+\frac {d^3 x (4 b c-3 a d)}{b^4}+\frac {x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac {d^4 x^3}{3 b^3} \]
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Rubi [A] time = 0.20, antiderivative size = 160, 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} \[ \frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}}+\frac {x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {d^3 x (4 b c-3 a d)}{b^4}+\frac {x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac {d^4 x^3}{3 b^3} \]
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 )^4}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac {d^3 (4 b c-3 a d)}{b^4}+\frac {d^4 x^2}{b^3}+\frac {b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{b^4 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {\int \frac {b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{\left (a+b x^2\right )^3} \, dx}{b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}-\frac {\int \frac {-(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+11 a^2 d^2\right )-24 a b d^2 (b c-a d)^2 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 160, normalized size = 1.00 \[ \frac {x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}}+\frac {d^3 x (4 b c-3 a d)}{b^4}+\frac {x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac {d^4 x^3}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 817, normalized size = 5.11 \[ \left [\frac {16 \, a^{3} b^{4} d^{4} x^{7} + 16 \, {\left (12 \, a^{3} b^{4} c d^{3} - 7 \, a^{4} b^{3} d^{4}\right )} x^{5} + 2 \, {\left (9 \, a b^{6} c^{4} + 12 \, a^{2} b^{5} c^{3} d - 90 \, a^{3} b^{4} c^{2} d^{2} + 300 \, a^{4} b^{3} c d^{3} - 175 \, a^{5} b^{2} d^{4}\right )} x^{3} - 3 \, {\left (3 \, a^{2} b^{4} c^{4} + 4 \, a^{3} b^{3} c^{3} d + 18 \, a^{4} b^{2} c^{2} d^{2} - 60 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (3 \, b^{6} c^{4} + 4 \, a b^{5} c^{3} d + 18 \, a^{2} b^{4} c^{2} d^{2} - 60 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, a b^{5} c^{4} + 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 60 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (5 \, a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d - 18 \, a^{4} b^{3} c^{2} d^{2} + 60 \, a^{5} b^{2} c d^{3} - 35 \, a^{6} b d^{4}\right )} x}{48 \, {\left (a^{3} b^{7} x^{4} + 2 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, \frac {8 \, a^{3} b^{4} d^{4} x^{7} + 8 \, {\left (12 \, a^{3} b^{4} c d^{3} - 7 \, a^{4} b^{3} d^{4}\right )} x^{5} + {\left (9 \, a b^{6} c^{4} + 12 \, a^{2} b^{5} c^{3} d - 90 \, a^{3} b^{4} c^{2} d^{2} + 300 \, a^{4} b^{3} c d^{3} - 175 \, a^{5} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b^{4} c^{4} + 4 \, a^{3} b^{3} c^{3} d + 18 \, a^{4} b^{2} c^{2} d^{2} - 60 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (3 \, b^{6} c^{4} + 4 \, a b^{5} c^{3} d + 18 \, a^{2} b^{4} c^{2} d^{2} - 60 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, a b^{5} c^{4} + 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 60 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (5 \, a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d - 18 \, a^{4} b^{3} c^{2} d^{2} + 60 \, a^{5} b^{2} c d^{3} - 35 \, a^{6} b d^{4}\right )} x}{24 \, {\left (a^{3} b^{7} x^{4} + 2 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 254, normalized size = 1.59 \[ \frac {{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{4}} + \frac {3 \, b^{5} c^{4} x^{3} + 4 \, a b^{4} c^{3} d x^{3} - 30 \, a^{2} b^{3} c^{2} d^{2} x^{3} + 36 \, a^{3} b^{2} c d^{3} x^{3} - 13 \, a^{4} b d^{4} x^{3} + 5 \, a b^{4} c^{4} x - 4 \, a^{2} b^{3} c^{3} d x - 18 \, a^{3} b^{2} c^{2} d^{2} x + 28 \, a^{4} b c d^{3} x - 11 \, a^{5} d^{4} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{4}} + \frac {b^{6} d^{4} x^{3} + 12 \, b^{6} c d^{3} x - 9 \, a b^{5} d^{4} x}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 367, normalized size = 2.29 \[ -\frac {13 a^{2} d^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {9 a c \,d^{3} x^{3}}{2 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {c^{3} d \,x^{3}}{2 \left (b \,x^{2}+a \right )^{2} a}+\frac {3 b \,c^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {15 c^{2} d^{2} x^{3}}{4 \left (b \,x^{2}+a \right )^{2} b}-\frac {11 a^{3} d^{4} x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {7 a^{2} c \,d^{3} x}{2 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {9 a \,c^{2} d^{2} x}{4 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {5 c^{4} x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {c^{3} d x}{2 \left (b \,x^{2}+a \right )^{2} b}+\frac {d^{4} x^{3}}{3 b^{3}}+\frac {35 a^{2} d^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {15 a c \,d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {c^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a b}+\frac {3 c^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {9 c^{2} d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{2}}-\frac {3 a \,d^{4} x}{b^{4}}+\frac {4 c \,d^{3} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 253, normalized size = 1.58 \[ \frac {{\left (3 \, b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 30 \, a^{2} b^{3} c^{2} d^{2} + 36 \, a^{3} b^{2} c d^{3} - 13 \, a^{4} b d^{4}\right )} x^{3} + {\left (5 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 28 \, a^{4} b c d^{3} - 11 \, a^{5} d^{4}\right )} x}{8 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} + \frac {b d^{4} x^{3} + 3 \, {\left (4 \, b c d^{3} - 3 \, a d^{4}\right )} x}{3 \, b^{4}} + \frac {{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 318, normalized size = 1.99 \[ \frac {d^4\,x^3}{3\,b^3}-x\,\left (\frac {3\,a\,d^4}{b^4}-\frac {4\,c\,d^3}{b^3}\right )-\frac {\frac {x\,\left (11\,a^4\,d^4-28\,a^3\,b\,c\,d^3+18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d-5\,b^4\,c^4\right )}{8\,a}-\frac {x^3\,\left (-13\,a^4\,b\,d^4+36\,a^3\,b^2\,c\,d^3-30\,a^2\,b^3\,c^2\,d^2+4\,a\,b^4\,c^3\,d+3\,b^5\,c^4\right )}{8\,a^2}}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (35\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{\sqrt {a}\,\left (35\,a^4\,d^4-60\,a^3\,b\,c\,d^3+18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+3\,b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (35\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^{5/2}\,b^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.81, size = 515, normalized size = 3.22 \[ x \left (- \frac {3 a d^{4}}{b^{4}} + \frac {4 c d^{3}}{b^{3}}\right ) - \frac {\sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- \frac {a^{3} b^{4} \sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log {\left (\frac {a^{3} b^{4} \sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} + \frac {x^{3} \left (- 13 a^{4} b d^{4} + 36 a^{3} b^{2} c d^{3} - 30 a^{2} b^{3} c^{2} d^{2} + 4 a b^{4} c^{3} d + 3 b^{5} c^{4}\right ) + x \left (- 11 a^{5} d^{4} + 28 a^{4} b c d^{3} - 18 a^{3} b^{2} c^{2} d^{2} - 4 a^{2} b^{3} c^{3} d + 5 a b^{4} c^{4}\right )}{8 a^{4} b^{4} + 16 a^{3} b^{5} x^{2} + 8 a^{2} b^{6} x^{4}} + \frac {d^{4} x^{3}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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