Optimal. Leaf size=281 \[ \frac {c^3 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x \sqrt {c+d x^2}}{1024 d^3}+\frac {c^2 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \sqrt {c+d x^2}}{512 d^2}+\frac {c \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac {\left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \left (c+d x^2\right )^{5/2}}{320 d^2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, 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 (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 {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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 285
Rule 327
Rule 470
Rule 475
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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.35, size = 225, normalized size = 0.80 \begin {gather*} \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 (5 b^2 c^2-24 a b c d+40 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{15360 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 353, normalized size = 1.26
method | result | size |
risch | \(\frac {x \left (1280 b^{2} d^{5} x^{10}+3072 a b \,d^{5} x^{8}+3200 b^{2} c \,d^{4} x^{8}+1920 a^{2} d^{5} x^{6}+8064 a b c \,d^{4} x^{6}+2160 b^{2} c^{2} d^{3} x^{6}+5440 a^{2} c \,d^{4} x^{4}+5952 a b \,c^{2} d^{3} x^{4}+40 b^{2} c^{3} d^{2} x^{4}+4720 a^{2} c^{2} d^{3} x^{2}+240 a b \,c^{3} d^{2} x^{2}-50 b^{2} c^{4} d \,x^{2}+600 a^{2} c^{3} d^{2}-360 a b \,c^{4} d +75 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{15360 d^{3}}-\frac {5 c^{4} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{128 d^{\frac {3}{2}}}+\frac {3 c^{5} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b}{128 d^{\frac {5}{2}}}-\frac {5 c^{6} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{1024 d^{\frac {7}{2}}}\) | \(272\) |
default | \(b^{2} \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{12 d}-\frac {5 c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )}{12 d}\right )+2 a b \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )+a^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )\) | \(353\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 361, normalized size = 1.28 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.01, size = 495, normalized size = 1.76 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.16, size = 265, normalized size = 0.94 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int 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.
[In]
[Out]
________________________________________________________________________________________