Optimal. Leaf size=202 \[ \frac {\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}}-\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac {x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt {c+d x^2}}+\frac {x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {463, 459, 288, 321, 217, 206} \[ \frac {x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt {c+d x^2}}-\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac {\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}}+\frac {x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 288
Rule 321
Rule 459
Rule 463
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {x^4 \left (-3 a^2 d^2+5 (b c-a d)^2-3 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^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {x^4}{\left (c+d x^2\right )^{3/2}} \, dx}{12 c d^2}\\ &=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{4 c d^3}\\ &=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 d^4}\\ &=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 d^4}\\ &=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 156, normalized size = 0.77 \[ \frac {\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{8 d^{9/2}}+\frac {x \left (-8 a^2 d^2 \left (3 c+4 d x^2\right )+8 a b d \left (15 c^2+20 c d x^2+3 d^2 x^4\right )-\left (b^2 \left (105 c^3+140 c^2 d x^2+21 c d^2 x^4-6 d^3 x^6\right )\right )\right )}{24 d^4 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 522, normalized size = 2.58 \[ \left [\frac {3 \, {\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (6 \, b^{2} d^{4} x^{7} - 3 \, {\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \, {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}}, -\frac {3 \, {\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (6 \, b^{2} d^{4} x^{7} - 3 \, {\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \, {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 190, normalized size = 0.94 \[ \frac {{\left ({\left (3 \, {\left (\frac {2 \, b^{2} x^{2}}{d} - \frac {7 \, b^{2} c^{2} d^{5} - 8 \, a b c d^{6}}{c d^{7}}\right )} x^{2} - \frac {4 \, {\left (35 \, b^{2} c^{3} d^{4} - 40 \, a b c^{2} d^{5} + 8 \, a^{2} c d^{6}\right )}}{c d^{7}}\right )} x^{2} - \frac {3 \, {\left (35 \, b^{2} c^{4} d^{3} - 40 \, a b c^{3} d^{4} + 8 \, a^{2} c^{2} d^{5}\right )}}{c d^{7}}\right )} x}{24 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{2} c^{2} - 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{8 \, d^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 255, normalized size = 1.26 \[ \frac {b^{2} x^{7}}{4 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {a b \,x^{5}}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} d}-\frac {7 b^{2} c \,x^{5}}{8 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{2}}-\frac {a^{2} x^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {5 a b c \,x^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{2}}-\frac {35 b^{2} c^{2} x^{3}}{24 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{3}}-\frac {a^{2} x}{\sqrt {d \,x^{2}+c}\, d^{2}}+\frac {5 a b c x}{\sqrt {d \,x^{2}+c}\, d^{3}}-\frac {35 b^{2} c^{2} x}{8 \sqrt {d \,x^{2}+c}\, d^{4}}+\frac {a^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{d^{\frac {5}{2}}}-\frac {5 a b c \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{d^{\frac {7}{2}}}+\frac {35 b^{2} c^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 296, normalized size = 1.47 \[ \frac {b^{2} x^{7}}{4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {7 \, b^{2} c x^{5}}{8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}} + \frac {a b x^{5}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {1}{3} \, a^{2} x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} - \frac {35 \, b^{2} c^{2} x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{24 \, d^{2}} + \frac {5 \, a b c x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{3 \, d} - \frac {35 \, b^{2} c^{2} x}{24 \, \sqrt {d x^{2} + c} d^{4}} + \frac {5 \, a b c x}{3 \, \sqrt {d x^{2} + c} d^{3}} - \frac {a^{2} x}{3 \, \sqrt {d x^{2} + c} d^{2}} + \frac {35 \, b^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {9}{2}}} - \frac {5 \, a b c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {7}{2}}} + \frac {a^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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