Optimal. Leaf size=375 \[ \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {474, 470,
327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}+\frac {\sqrt {x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac {x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{9/2}}{9 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 217
Rule 327
Rule 335
Rule 470
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{7/2} \left (\frac {1}{2} (3 b c-5 a d) (3 b c-a d)-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{7/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{3/2}}{c+d x^2} \, dx}{4 d^3}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{9/2}}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{9/2}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{17/4}}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}\\ &=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.63, size = 255, normalized size = 0.68 \begin {gather*} \frac {\frac {4 \sqrt [4]{d} \sqrt {x} \left (45 a^2 d^2 \left (5 c+4 d x^2\right )+18 a b d \left (-45 c^2-36 c d x^2+4 d^2 x^4\right )+b^2 \left (585 c^3+468 c^2 d x^2-52 c d^2 x^4+20 d^3 x^6\right )\right )}{c+d x^2}+45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{360 d^{17/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 240, normalized size = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} x^{\frac {9}{2}} d^{2}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
default | \(\frac {\frac {2 b^{2} x^{\frac {9}{2}} d^{2}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
risch | \(\frac {2 \left (5 b^{2} d^{2} x^{4}+18 a b \,d^{2} x^{2}-18 b^{2} c d \,x^{2}+45 a^{2} d^{2}-180 a b c d +135 b^{2} c^{2}\right ) \sqrt {x}}{45 d^{4}}+\frac {c \sqrt {x}\, a^{2}}{2 d^{2} \left (d \,x^{2}+c \right )}-\frac {c^{2} \sqrt {x}\, a b}{d^{3} \left (d \,x^{2}+c \right )}+\frac {c^{3} \sqrt {x}\, b^{2}}{2 d^{4} \left (d \,x^{2}+c \right )}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{8 d^{2}}+\frac {9 c \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{4 d^{3}}-\frac {13 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{8 d^{4}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{8 d^{2}}+\frac {9 c \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{4 d^{3}}-\frac {13 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2}}{8 d^{4}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{16 d^{2}}+\frac {9 c \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{8 d^{3}}-\frac {13 c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2}}{16 d^{4}}\) | \(556\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 377, normalized size = 1.01 \begin {gather*} \frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}}{2 \, {\left (d^{5} x^{2} + c d^{4}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{16 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 18 \, {\left (b^{2} c d - a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1373 vs.
\(2 (287) = 574\).
time = 0.52, size = 1373, normalized size = 3.66 \begin {gather*} -\frac {180 \, {\left (d^{5} x^{2} + c d^{4}\right )} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d^{8} \sqrt {-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}} + {\left (169 \, b^{4} c^{4} - 468 \, a b^{3} c^{3} d + 454 \, a^{2} b^{2} c^{2} d^{2} - 180 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4}\right )} x} d^{13} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {3}{4}} - {\left (13 \, b^{2} c^{2} d^{13} - 18 \, a b c d^{14} + 5 \, a^{2} d^{15}\right )} \sqrt {x} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {3}{4}}}{28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}\right ) + 45 \, {\left (d^{5} x^{2} + c d^{4}\right )} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {1}{4}} \log \left (d^{4} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {1}{4}} + {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - 45 \, {\left (d^{5} x^{2} + c d^{4}\right )} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {1}{4}} \log \left (-d^{4} \left (-\frac {28561 \, b^{8} c^{9} - 158184 \, a b^{7} c^{8} d + 372476 \, a^{2} b^{6} c^{7} d^{2} - 485784 \, a^{3} b^{5} c^{6} d^{3} + 383046 \, a^{4} b^{4} c^{5} d^{4} - 186840 \, a^{5} b^{3} c^{4} d^{5} + 55100 \, a^{6} b^{2} c^{3} d^{6} - 9000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8}}{d^{17}}\right )^{\frac {1}{4}} + {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (20 \, b^{2} d^{3} x^{6} + 585 \, b^{2} c^{3} - 810 \, a b c^{2} d + 225 \, a^{2} c d^{2} - 4 \, {\left (13 \, b^{2} c d^{2} - 18 \, a b d^{3}\right )} x^{4} + 36 \, {\left (13 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {x}}{360 \, {\left (d^{5} x^{2} + c 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 = 0.63, size = 440, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {b^{2} c^{3} \sqrt {x} - 2 \, a b c^{2} d \sqrt {x} + a^{2} c d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{16} x^{\frac {9}{2}} - 18 \, b^{2} c d^{15} x^{\frac {5}{2}} + 18 \, a b d^{16} x^{\frac {5}{2}} + 135 \, b^{2} c^{2} d^{14} \sqrt {x} - 180 \, a b c d^{15} \sqrt {x} + 45 \, a^{2} d^{16} \sqrt {x}\right )}}{45 \, d^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.24, size = 1367, normalized size = 3.65 \begin {gather*} \sqrt {x}\,\left (\frac {2\,a^2}{d^2}+\frac {2\,c\,\left (\frac {4\,b^2\,c}{d^3}-\frac {4\,a\,b}{d^2}\right )}{d}-\frac {2\,b^2\,c^2}{d^4}\right )-x^{5/2}\,\left (\frac {4\,b^2\,c}{5\,d^3}-\frac {4\,a\,b}{5\,d^2}\right )+\frac {\sqrt {x}\,\left (\frac {a^2\,c\,d^2}{2}-a\,b\,c^2\,d+\frac {b^2\,c^3}{2}\right )}{d^5\,x^2+c\,d^4}+\frac {2\,b^2\,x^{9/2}}{9\,d^2}+\frac {{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}+\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,1{}\mathrm {i}}{8\,d^{17/4}}+\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}-\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,1{}\mathrm {i}}{8\,d^{17/4}}}{\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}+\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )}{8\,d^{17/4}}-\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}-\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )}{8\,d^{17/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,1{}\mathrm {i}}{4\,d^{17/4}}-\frac {{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}-\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )\,1{}\mathrm {i}}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )}{8\,d^{17/4}}+\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}+\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )\,1{}\mathrm {i}}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )}{8\,d^{17/4}}}{\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}-\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )\,1{}\mathrm {i}}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,1{}\mathrm {i}}{8\,d^{17/4}}-\frac {{\left (-c\right )}^{1/4}\,\left (\frac {\sqrt {x}\,\left (25\,a^4\,c^2\,d^4-180\,a^3\,b\,c^3\,d^3+454\,a^2\,b^2\,c^4\,d^2-468\,a\,b^3\,c^5\,d+169\,b^4\,c^6\right )}{d^5}+\frac {{\left (-c\right )}^{1/4}\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,\left (5\,a^2\,c^2\,d^2-18\,a\,b\,c^3\,d+13\,b^2\,c^4\right )\,1{}\mathrm {i}}{d^{21/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )\,1{}\mathrm {i}}{8\,d^{17/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-13\,b\,c\right )}{4\,d^{17/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________