Optimal. Leaf size=147 \[ \frac {d x \left (105 a^2 d^2-190 a b c d+81 b^2 c^2\right )}{30 b^4}+\frac {(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{9/2}}+\frac {d x \left (c+d x^2\right ) (33 b c-35 a d)}{30 b^3}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {7 d x \left (c+d x^2\right )^2}{10 b^2} \]
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Rubi [A] time = 0.19, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {467, 528, 388, 205} \begin {gather*} \frac {d x \left (105 a^2 d^2-190 a b c d+81 b^2 c^2\right )}{30 b^4}+\frac {d x \left (c+d x^2\right ) (33 b c-35 a d)}{30 b^3}+\frac {(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{9/2}}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {7 d x \left (c+d x^2\right )^2}{10 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 467
Rule 528
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (c+7 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac {7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (5 b c-7 a d)+d (33 b c-35 a d) x^2\right )}{a+b x^2} \, dx}{10 b^2}\\ &=\frac {d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac {7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\int \frac {c \left (15 b^2 c^2-54 a b c d+35 a^2 d^2\right )+d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x^2}{a+b x^2} \, dx}{30 b^3}\\ &=\frac {d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x}{30 b^4}+\frac {d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac {7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-7 a d) (b c-a d)^2\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^4}\\ &=\frac {d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x}{30 b^4}+\frac {d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac {7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac {x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 125, normalized size = 0.85 \begin {gather*} \frac {(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{9/2}}-\frac {x (b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 d x (b c-a d)^2}{b^4}+\frac {d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac {d^3 x^5}{5 b^2} \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 {x^2 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.96, size = 508, normalized size = 3.46 \begin {gather*} \left [\frac {12 \, a b^{4} d^{3} x^{7} + 4 \, {\left (15 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{5} + 20 \, {\left (9 \, a b^{4} c^{2} d - 15 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} 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 ) - 30 \, {\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 7 \, a^{4} b d^{3}\right )} x}{60 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac {6 \, a b^{4} d^{3} x^{7} + 2 \, {\left (15 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (9 \, a b^{4} c^{2} d - 15 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 15 \, {\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 7 \, a^{4} b d^{3}\right )} x}{30 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 184, normalized size = 1.25 \begin {gather*} \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} - \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} d^{3} x^{5} + 15 \, b^{8} c d^{2} x^{3} - 10 \, a b^{7} d^{3} x^{3} + 45 \, b^{8} c^{2} d x - 90 \, a b^{7} c d^{2} x + 45 \, a^{2} b^{6} d^{3} x}{15 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 247, normalized size = 1.68 \begin {gather*} \frac {d^{3} x^{5}}{5 b^{2}}-\frac {2 a \,d^{3} x^{3}}{3 b^{3}}+\frac {c \,d^{2} x^{3}}{b^{2}}+\frac {a^{3} d^{3} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {3 a^{2} c \,d^{2} x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {15 a^{2} c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {3 a \,c^{2} d x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {9 a \,c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {c^{3} x}{2 \left (b \,x^{2}+a \right ) b}+\frac {c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {3 a^{2} d^{3} x}{b^{4}}-\frac {6 a c \,d^{2} x}{b^{3}}+\frac {3 c^{2} d x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.29, size = 176, normalized size = 1.20 \begin {gather*} -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} d^{3} x^{5} + 5 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{3} + 45 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{15 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 232, normalized size = 1.58 \begin {gather*} x\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )-x^3\,\left (\frac {2\,a\,d^3}{3\,b^3}-\frac {c\,d^2}{b^2}\right )+\frac {x\,\left (\frac {a^3\,d^3}{2}-\frac {3\,a^2\,b\,c\,d^2}{2}+\frac {3\,a\,b^2\,c^2\,d}{2}-\frac {b^3\,c^3}{2}\right )}{b^5\,x^2+a\,b^4}+\frac {d^3\,x^5}{5\,b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d-b\,c\right )}{\sqrt {a}\,\left (7\,a^3\,d^3-15\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d-b\,c\right )}{2\,\sqrt {a}\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.22, size = 338, normalized size = 2.30 \begin {gather*} x^{3} \left (- \frac {2 a d^{3}}{3 b^{3}} + \frac {c d^{2}}{b^{2}}\right ) + x \left (\frac {3 a^{2} d^{3}}{b^{4}} - \frac {6 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{b^{2}}\right ) + \frac {x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \log {\left (- \frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right )}{7 a^{3} d^{3} - 15 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \log {\left (\frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right )}{7 a^{3} d^{3} - 15 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{4} + \frac {d^{3} x^{5}}{5 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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