Optimal. Leaf size=98 \[ \frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac {b^2 (b c-3 a d) x^3}{3 d^2}+\frac {b^3 x^5}{5 d}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {398, 211}
\begin {gather*} \frac {b x \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}-\frac {(b c-a d)^3 \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}}-\frac {b^2 x^3 (b c-3 a d)}{3 d^2}+\frac {b^3 x^5}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 398
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx &=\int \left (\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{d^3}-\frac {b^2 (b c-3 a d) x^2}{d^2}+\frac {b^3 x^4}{d}+\frac {-b^3 c^3+3 a b^2 c^2 d-3 a^2 b c d^2+a^3 d^3}{d^3 \left (c+d x^2\right )}\right ) \, dx\\ &=\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac {b^2 (b c-3 a d) x^3}{3 d^2}+\frac {b^3 x^5}{5 d}-\frac {(b c-a d)^3 \int \frac {1}{c+d x^2} \, dx}{d^3}\\ &=\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac {b^2 (b c-3 a d) x^3}{3 d^2}+\frac {b^3 x^5}{5 d}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 93, normalized size = 0.95 \begin {gather*} \frac {b x \left (45 a^2 d^2+15 a b d \left (-3 c+d x^2\right )+b^2 \left (15 c^2-5 c d x^2+3 d^2 x^4\right )\right )}{15 d^3}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 116, normalized size = 1.18
method | result | size |
default | \(\frac {b \left (\frac {1}{5} b^{2} x^{5} d^{2}+a b \,d^{2} x^{3}-\frac {1}{3} b^{2} c d \,x^{3}+3 a^{2} d^{2} x -3 a b c d x +b^{2} c^{2} x \right )}{d^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{d^{3} \sqrt {c d}}\) | \(116\) |
risch | \(\frac {b^{3} x^{5}}{5 d}+\frac {b^{2} a \,x^{3}}{d}-\frac {b^{3} c \,x^{3}}{3 d^{2}}+\frac {3 b \,a^{2} x}{d}-\frac {3 b^{2} a c x}{d^{2}}+\frac {b^{3} c^{2} x}{d^{3}}-\frac {\ln \left (d x +\sqrt {-c d}\right ) a^{3}}{2 \sqrt {-c d}}+\frac {3 \ln \left (d x +\sqrt {-c d}\right ) a^{2} b c}{2 d \sqrt {-c d}}-\frac {3 \ln \left (d x +\sqrt {-c d}\right ) a \,b^{2} c^{2}}{2 d^{2} \sqrt {-c d}}+\frac {\ln \left (d x +\sqrt {-c d}\right ) b^{3} c^{3}}{2 d^{3} \sqrt {-c d}}+\frac {\ln \left (-d x +\sqrt {-c d}\right ) a^{3}}{2 \sqrt {-c d}}-\frac {3 \ln \left (-d x +\sqrt {-c d}\right ) a^{2} b c}{2 d \sqrt {-c d}}+\frac {3 \ln \left (-d x +\sqrt {-c d}\right ) a \,b^{2} c^{2}}{2 d^{2} \sqrt {-c d}}-\frac {\ln \left (-d x +\sqrt {-c d}\right ) b^{3} c^{3}}{2 d^{3} \sqrt {-c d}}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 122, normalized size = 1.24 \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 )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{2} x^{5} - 5 \, {\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{3} + 15 \, {\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{15 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 290, normalized size = 2.96 \begin {gather*} \left [\frac {6 \, b^{3} c d^{3} x^{5} - 10 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 30 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{30 \, c d^{4}}, \frac {3 \, b^{3} c d^{3} x^{5} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} - 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + 15 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{15 \, c d^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs.
\(2 (92) = 184\).
time = 0.31, size = 238, normalized size = 2.43 \begin {gather*} \frac {b^{3} x^{5}}{5 d} + x^{3} \left (\frac {a b^{2}}{d} - \frac {b^{3} c}{3 d^{2}}\right ) + x \left (\frac {3 a^{2} b}{d} - \frac {3 a b^{2} c}{d^{2}} + \frac {b^{3} c^{2}}{d^{3}}\right ) - \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (- \frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (\frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 130, normalized size = 1.33 \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 )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{4} x^{5} - 5 \, b^{3} c d^{3} x^{3} + 15 \, a b^{2} d^{4} x^{3} + 15 \, b^{3} c^{2} d^{2} x - 45 \, a b^{2} c d^{3} x + 45 \, a^{2} b d^{4} x}{15 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.87, size = 145, normalized size = 1.48 \begin {gather*} x^3\,\left (\frac {a\,b^2}{d}-\frac {b^3\,c}{3\,d^2}\right )+x\,\left (\frac {3\,a^2\,b}{d}-\frac {c\,\left (\frac {3\,a\,b^2}{d}-\frac {b^3\,c}{d^2}\right )}{d}\right )+\frac {b^3\,x^5}{5\,d}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,d^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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