Optimal. Leaf size=281 \[ -\frac {c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{7/2}}+\frac {c^3 x \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}+\frac {c^2 x^3 \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac {x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{320 d^2}+\frac {c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac {b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]
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Rubi [A] time = 0.26, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {464, 459, 279, 321, 217, 206} \[ \frac {c^2 x^3 \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac {c^3 x \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}-\frac {c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{7/2}}+\frac {1}{320} x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )+\frac {c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac {b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 459
Rule 464
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx &=\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {\int x^2 \left (c+d x^2\right )^{5/2} \left (12 a^2 d-b (5 b c-24 a d) x^2\right ) \, dx}{12 d}\\ &=-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{40} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) \int x^2 \left (c+d x^2\right )^{5/2} \, dx\\ &=\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{64} \left (c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{128} \left (c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \sqrt {c+d x^2} \, dx\\ &=\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{512} \left (c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx\\ &=\frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {\left (c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{1024 d}\\ &=\frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {\left (c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{1024 d}\\ &=\frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 226, normalized size = 0.80 \[ \frac {\sqrt {d} x \sqrt {c+d x^2} \left (40 a^2 d^2 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )+24 a b d \left (-15 c^4+10 c^3 d x^2+248 c^2 d^2 x^4+336 c d^3 x^6+128 d^4 x^8\right )+5 b^2 \left (15 c^5-10 c^4 d x^2+8 c^3 d^2 x^4+432 c^2 d^3 x^6+640 c d^4 x^8+256 d^5 x^{10}\right )\right )-15 c^4 \left (40 a^2 d^2-24 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{15360 d^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 495, normalized size = 1.76 \[ \left [\frac {15 \, {\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \, {\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \, {\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \, {\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{30720 \, d^{4}}, \frac {15 \, {\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \, {\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \, {\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \, {\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{15360 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 265, normalized size = 0.94 \[ \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d^{2} x^{2} + \frac {25 \, b^{2} c d^{11} + 24 \, a b d^{12}}{d^{10}}\right )} x^{2} + \frac {3 \, {\left (45 \, b^{2} c^{2} d^{10} + 168 \, a b c d^{11} + 40 \, a^{2} d^{12}\right )}}{d^{10}}\right )} x^{2} + \frac {5 \, b^{2} c^{3} d^{9} + 744 \, a b c^{2} d^{10} + 680 \, a^{2} c d^{11}}{d^{10}}\right )} x^{2} - \frac {5 \, {\left (5 \, b^{2} c^{4} d^{8} - 24 \, a b c^{3} d^{9} - 472 \, a^{2} c^{2} d^{10}\right )}}{d^{10}}\right )} x^{2} + \frac {15 \, {\left (5 \, b^{2} c^{5} d^{7} - 24 \, a b c^{4} d^{8} + 40 \, a^{2} c^{3} d^{9}\right )}}{d^{10}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{1024 \, d^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 383, normalized size = 1.36 \[ \frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} b^{2} x^{5}}{12 d}-\frac {5 a^{2} c^{4} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {3}{2}}}+\frac {3 a b \,c^{5} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {5}{2}}}-\frac {5 b^{2} c^{6} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{1024 d^{\frac {7}{2}}}-\frac {5 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} x}{128 d}+\frac {3 \sqrt {d \,x^{2}+c}\, a b \,c^{4} x}{128 d^{2}}-\frac {5 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} x}{1024 d^{3}}-\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} c^{2} x}{192 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,c^{3} x}{64 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a b \,x^{3}}{5 d}-\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c^{4} x}{1536 d^{3}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} b^{2} c \,x^{3}}{24 d^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2} c x}{48 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a b \,c^{2} x}{80 d^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c^{3} x}{384 d^{3}}+\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2} x}{8 d}-\frac {3 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a b c x}{40 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} b^{2} c^{2} x}{64 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 361, normalized size = 1.28 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{5}}{12 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{3}}{24 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{3}}{5 \, d} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} x}{64 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x}{384 \, d^{3}} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} x}{1536 \, d^{3}} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{5} x}{1024 \, d^{3}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c x}{40 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x}{80 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a b c^{4} x}{128 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} x}{8 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x}{48 \, d} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} x}{192 \, d} - \frac {5 \, \sqrt {d x^{2} + c} a^{2} c^{3} x}{128 \, d} - \frac {5 \, b^{2} c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{1024 \, d^{\frac {7}{2}}} + \frac {3 \, a b c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} - \frac {5 \, a^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {3}{2}}} \]
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 \[ \int x^2\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 96.02, size = 602, normalized size = 2.14 \[ \frac {5 a^{2} c^{\frac {7}{2}} x}{128 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {133 a^{2} c^{\frac {5}{2}} x^{3}}{384 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {127 a^{2} c^{\frac {3}{2}} d x^{5}}{192 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {23 a^{2} \sqrt {c} d^{2} x^{7}}{48 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {5 a^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {3}{2}}} + \frac {a^{2} d^{3} x^{9}}{8 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {3 a b c^{\frac {9}{2}} x}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{\frac {7}{2}} x^{3}}{128 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {129 a b c^{\frac {5}{2}} x^{5}}{320 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {73 a b c^{\frac {3}{2}} d x^{7}}{80 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {29 a b \sqrt {c} d^{2} x^{9}}{40 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b c^{5} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {5}{2}}} + \frac {a b d^{3} x^{11}}{5 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {11}{2}} x}{1024 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {9}{2}} x^{3}}{3072 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {7}{2}} x^{5}}{1536 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {55 b^{2} c^{\frac {5}{2}} x^{7}}{384 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {67 b^{2} c^{\frac {3}{2}} d x^{9}}{192 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} \sqrt {c} d^{2} x^{11}}{24 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {5 b^{2} c^{6} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{1024 d^{\frac {7}{2}}} + \frac {b^{2} d^{3} x^{13}}{12 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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