Optimal. Leaf size=121 \[ -\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}+\frac {2 b x (b c-a d)}{d^3 \sqrt {c+d x^2}}+\frac {x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3} \]
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Rubi [A] time = 0.11, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {463, 455, 388, 217, 206} \begin {gather*} \frac {x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b x (b c-a d)}{d^3 \sqrt {c+d x^2}}-\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
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 )^{5/2}} \, dx &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (3 b c (b c-2 a d)-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {\int \frac {-6 b c d (b c-a d)+3 b^2 c d^2 x^2}{\sqrt {c+d x^2}} \, dx}{3 c d^4}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 d^3}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 d^3}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 118, normalized size = 0.98 \begin {gather*} \frac {x \left (2 a^2 d^3 x^2-4 a b c d \left (3 c+4 d x^2\right )+b^2 c \left (15 c^2+20 c d x^2+3 d^2 x^4\right )\right )}{6 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {b (4 a d-5 b c) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{2 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 129, normalized size = 1.07 \begin {gather*} \frac {2 a^2 d^3 x^3-12 a b c^2 d x-16 a b c d^2 x^3+15 b^2 c^3 x+20 b^2 c^2 d x^3+3 b^2 c d^2 x^5}{6 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (5 b^2 c-4 a b d\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{2 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 409, normalized size = 3.38 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 130, normalized size = 1.07 \begin {gather*} \frac {{\left ({\left (\frac {3 \, b^{2} x^{2}}{d} + \frac {2 \, {\left (10 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + a^{2} d^{5}\right )}}{c d^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{2} - 4 \, a b c^{2} d^{3}\right )}}{c d^{5}}\right )} x}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - 4 \, a b d\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{2 \, d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 185, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{5}}{2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}-\frac {2 a b \,x^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {5 b^{2} c \,x^{3}}{6 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{2}}-\frac {a^{2} x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {a^{2} x}{3 \sqrt {d \,x^{2}+c}\, c d}-\frac {2 a b x}{\sqrt {d \,x^{2}+c}\, d^{2}}+\frac {5 b^{2} c x}{2 \sqrt {d \,x^{2}+c}\, d^{3}}+\frac {2 a b \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{d^{\frac {5}{2}}}-\frac {5 b^{2} c \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 211, normalized size = 1.74 \begin {gather*} \frac {b^{2} x^{5}}{2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {2}{3} \, a b x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} + \frac {5 \, b^{2} c x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{6 \, d} + \frac {5 \, b^{2} c x}{6 \, \sqrt {d x^{2} + c} d^{3}} - \frac {2 \, a b x}{3 \, \sqrt {d x^{2} + c} d^{2}} - \frac {a^{2} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {a^{2} x}{3 \, \sqrt {d x^{2} + c} c d} - \frac {5 \, b^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, d^{\frac {7}{2}}} + \frac {2 \, a b \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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