Optimal. Leaf size=281 \[ -\frac {c^3 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x \sqrt {c+d x^2}}{1024 d^4}+\frac {c^2 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^3 \sqrt {c+d x^2}}{1536 d^3}+\frac {c \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^5 \sqrt {c+d x^2}}{384 d^2}+\frac {\left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^5 \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{9/2}} \]
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Rubi [A]
time = 0.18, 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 = {475, 470, 285,
327, 223, 212} \begin {gather*} \frac {c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{9/2}}-\frac {c^3 x \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac {c^2 x^3 \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac {1}{192} x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )+\frac {c x^5 \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac {b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 470
Rule 475
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\int x^4 \left (c+d x^2\right )^{3/2} \left (12 a^2 d-b (7 b c-24 a d) x^2\right ) \, dx}{12 d}\\ &=-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{24} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) \int x^4 \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{64} \left (c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int x^4 \sqrt {c+d x^2} \, dx\\ &=\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{384} \left (c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^4}{\sqrt {c+d x^2}} \, dx\\ &=\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}-\frac {\left (c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{512 d}\\ &=-\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\left (c^4 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{1024 d^2}\\ &=-\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\left (c^4 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{1024 d^2}\\ &=-\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {c^4 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 224, normalized size = 0.80 \begin {gather*} \frac {\sqrt {d} x \sqrt {c+d x^2} \left (120 a^2 d^2 \left (-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )+24 a b d \left (15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right )+b^2 \left (-105 c^5+70 c^4 d x^2-56 c^3 d^2 x^4+48 c^2 d^3 x^6+1664 c d^4 x^8+1280 d^5 x^{10}\right )\right )-15 c^4 \left (7 b^2 c^2-24 a b c d+24 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{15360 d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 377, normalized size = 1.34
method | result | size |
risch | \(-\frac {x \left (-1280 b^{2} d^{5} x^{10}-3072 a b \,d^{5} x^{8}-1664 b^{2} c \,d^{4} x^{8}-1920 a^{2} d^{5} x^{6}-4224 a b c \,d^{4} x^{6}-48 b^{2} c^{2} d^{3} x^{6}-2880 a^{2} c \,d^{4} x^{4}-192 a b \,c^{2} d^{3} x^{4}+56 b^{2} c^{3} d^{2} x^{4}-240 a^{2} c^{2} d^{3} x^{2}+240 a b \,c^{3} d^{2} x^{2}-70 b^{2} c^{4} d \,x^{2}+360 a^{2} c^{3} d^{2}-360 a b \,c^{4} d +105 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{15360 d^{4}}+\frac {3 c^{4} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{128 d^{\frac {5}{2}}}-\frac {3 c^{5} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b}{128 d^{\frac {7}{2}}}+\frac {7 c^{6} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{1024 d^{\frac {9}{2}}}\) | \(272\) |
default | \(b^{2} \left (\frac {x^{7} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{12 d}-\frac {7 c \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{10 d}-\frac {c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )}{2 d}\right )}{12 d}\right )+2 a b \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{10 d}-\frac {c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )}{2 d}\right )+a^{2} \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )\) | \(377\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 367, normalized size = 1.31 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{7}}{12 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{5}}{120 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{5}}{5 \, d} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} x^{3}}{192 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c x^{3}}{8 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} x^{3}}{8 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x}{384 \, d^{4}} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} x}{1536 \, d^{4}} + \frac {7 \, \sqrt {d x^{2} + c} b^{2} c^{5} x}{1024 \, d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x}{16 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} x}{64 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a b c^{4} x}{128 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x}{16 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a^{2} c^{3} x}{128 \, d^{2}} + \frac {7 \, b^{2} c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{1024 \, d^{\frac {9}{2}}} - \frac {3 \, a b c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {7}{2}}} + \frac {3 \, a^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 494, normalized size = 1.76 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, 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 (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{30720 \, d^{5}}, -\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, 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 (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{15360 \, d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 263, normalized size = 0.94 \begin {gather*} \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d x^{2} + \frac {13 \, b^{2} c d^{10} + 24 \, a b d^{11}}{d^{10}}\right )} x^{2} + \frac {3 \, {\left (b^{2} c^{2} d^{9} + 88 \, a b c d^{10} + 40 \, a^{2} d^{11}\right )}}{d^{10}}\right )} x^{2} - \frac {7 \, b^{2} c^{3} d^{8} - 24 \, a b c^{2} d^{9} - 360 \, a^{2} c d^{10}}{d^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, b^{2} c^{4} d^{7} - 24 \, a b c^{3} d^{8} + 24 \, a^{2} c^{2} d^{9}\right )}}{d^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, b^{2} c^{5} d^{6} - 24 \, a b c^{4} d^{7} + 24 \, a^{2} c^{3} d^{8}\right )}}{d^{10}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{1024 \, d^{\frac {9}{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 = -0.00 \begin {gather*} \int x^4\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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