Optimal. Leaf size=163 \[ -\frac {\left (13 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{4 c d^4}+\frac {b^2 x^3}{3 d^3}+\frac {(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{9/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 163, 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 = {474, 466, 1167,
211} \begin {gather*} \frac {\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{9/2}}-\frac {x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}-\frac {x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac {x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {b^2 x^3}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 466
Rule 474
Rule 1167
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac {(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^4 \left (-4 a^2 d^2+5 (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^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac {\int \frac {c d (b c-a d) (9 b c-a d)-2 d^2 (b c-a d) (9 b c-a d) x^2+8 b^2 c d^3 x^4}{c+d x^2} \, dx}{8 c d^5}\\ &=\frac {(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac {\int \left (-2 d \left (13 b^2 c^2-10 a b c d+a^2 d^2\right )+8 b^2 c d^2 x^2+\frac {35 b^2 c^3 d-30 a b c^2 d^2+3 a^2 c d^3}{c+d x^2}\right ) \, dx}{8 c d^5}\\ &=-\frac {\left (13 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{4 c d^4}+\frac {b^2 x^3}{3 d^3}+\frac {(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 d^4}\\ &=-\frac {\left (13 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{4 c d^4}+\frac {b^2 x^3}{3 d^3}+\frac {(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 148, normalized size = 0.91 \begin {gather*} -\frac {b (3 b c-2 a d) x}{d^4}+\frac {b^2 x^3}{3 d^3}+\frac {c (b c-a d)^2 x}{4 d^4 \left (c+d x^2\right )^2}-\frac {\left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) x}{8 d^4 \left (c+d x^2\right )}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 137, normalized size = 0.84
method | result | size |
default | \(\frac {b \left (\frac {1}{3} b d \,x^{3}+2 a d x -3 b c x \right )}{d^{4}}+\frac {\frac {\left (-\frac {5}{8} a^{2} d^{3}+\frac {9}{4} a b c \,d^{2}-\frac {13}{8} b^{2} c^{2} d \right ) x^{3}-\frac {c \left (3 a^{2} d^{2}-14 a b c d +11 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-30 a b c d +35 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}}{d^{4}}\) | \(137\) |
risch | \(\frac {b^{2} x^{3}}{3 d^{3}}+\frac {2 b a x}{d^{3}}-\frac {3 b^{2} c x}{d^{4}}+\frac {\left (-\frac {5}{8} a^{2} d^{3}+\frac {9}{4} a b c \,d^{2}-\frac {13}{8} b^{2} c^{2} d \right ) x^{3}-\frac {c \left (3 a^{2} d^{2}-14 a b c d +11 b^{2} c^{2}\right ) x}{8}}{d^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {3 \ln \left (d x +\sqrt {-c d}\right ) a^{2}}{16 d^{2} \sqrt {-c d}}+\frac {15 \ln \left (d x +\sqrt {-c d}\right ) a b c}{8 d^{3} \sqrt {-c d}}-\frac {35 \ln \left (d x +\sqrt {-c d}\right ) b^{2} c^{2}}{16 d^{4} \sqrt {-c d}}+\frac {3 \ln \left (-d x +\sqrt {-c d}\right ) a^{2}}{16 d^{2} \sqrt {-c d}}-\frac {15 \ln \left (-d x +\sqrt {-c d}\right ) a b c}{8 d^{3} \sqrt {-c d}}+\frac {35 \ln \left (-d x +\sqrt {-c d}\right ) b^{2} c^{2}}{16 d^{4} \sqrt {-c d}}\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 159, normalized size = 0.98 \begin {gather*} -\frac {{\left (13 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{3} + {\left (11 \, b^{2} c^{3} - 14 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x}{8 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} + \frac {{\left (35 \, b^{2} c^{2} - 30 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} d^{4}} + \frac {b^{2} d x^{3} - 3 \, {\left (3 \, b^{2} c - 2 \, a b d\right )} x}{3 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 522, normalized size = 3.20 \begin {gather*} \left [\frac {16 \, b^{2} c d^{4} x^{7} - 16 \, {\left (7 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4}\right )} x^{5} - 10 \, {\left (35 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 3 \, a^{2} c d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, 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 ) - 6 \, {\left (35 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3}\right )} x}{48 \, {\left (c d^{7} x^{4} + 2 \, c^{2} d^{6} x^{2} + c^{3} d^{5}\right )}}, \frac {8 \, b^{2} c d^{4} x^{7} - 8 \, {\left (7 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4}\right )} x^{5} - 5 \, {\left (35 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 3 \, a^{2} c d^{4}\right )} x^{3} + 3 \, {\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - 3 \, {\left (35 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3}\right )} x}{24 \, {\left (c d^{7} x^{4} + 2 \, c^{2} d^{6} x^{2} + c^{3} d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.20, size = 240, normalized size = 1.47 \begin {gather*} \frac {b^{2} x^{3}}{3 d^{3}} + x \left (\frac {2 a b}{d^{3}} - \frac {3 b^{2} c}{d^{4}}\right ) - \frac {\sqrt {- \frac {1}{c d^{9}}} \cdot \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log {\left (- c d^{4} \sqrt {- \frac {1}{c d^{9}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c d^{9}}} \cdot \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log {\left (c d^{4} \sqrt {- \frac {1}{c d^{9}}} + x \right )}}{16} + \frac {x^{3} \left (- 5 a^{2} d^{3} + 18 a b c d^{2} - 13 b^{2} c^{2} d\right ) + x \left (- 3 a^{2} c d^{2} + 14 a b c^{2} d - 11 b^{2} c^{3}\right )}{8 c^{2} d^{4} + 16 c d^{5} x^{2} + 8 d^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 154, normalized size = 0.94 \begin {gather*} \frac {{\left (35 \, b^{2} c^{2} - 30 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} d^{4}} - \frac {13 \, b^{2} c^{2} d x^{3} - 18 \, a b c d^{2} x^{3} + 5 \, a^{2} d^{3} x^{3} + 11 \, b^{2} c^{3} x - 14 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} d^{4}} + \frac {b^{2} d^{6} x^{3} - 9 \, b^{2} c d^{5} x + 6 \, a b d^{6} x}{3 \, d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 159, normalized size = 0.98 \begin {gather*} \frac {b^2\,x^3}{3\,d^3}-\frac {\left (\frac {5\,a^2\,d^3}{8}-\frac {9\,a\,b\,c\,d^2}{4}+\frac {13\,b^2\,c^2\,d}{8}\right )\,x^3+\left (\frac {3\,a^2\,c\,d^2}{8}-\frac {7\,a\,b\,c^2\,d}{4}+\frac {11\,b^2\,c^3}{8}\right )\,x}{c^2\,d^4+2\,c\,d^5\,x^2+d^6\,x^4}-x\,\left (\frac {3\,b^2\,c}{d^4}-\frac {2\,a\,b}{d^3}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (3\,a^2\,d^2-30\,a\,b\,c\,d+35\,b^2\,c^2\right )}{8\,\sqrt {c}\,d^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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