Optimal. Leaf size=305 \[ -\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}} \]
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Rubi [A]
time = 0.18, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 472, 217,
1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}-\frac {2 c^3}{11 a x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 472
Rule 477
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^{12} \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {c^3}{a x^{12}}+\frac {c^2 (-b c+3 a d)}{a^2 x^8}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^4}+\frac {(-b c+a d)^3}{a^3 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{7/2}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{7/2}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{7/2} \sqrt {b}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{7/2} \sqrt {b}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 194, normalized size = 0.64 \begin {gather*} \frac {-\frac {4 a^{3/4} c \left (77 b^2 c^2 x^4-33 a b c x^2 \left (c+7 d x^2\right )+3 a^2 \left (7 c^2+33 c d x^2+77 d^2 x^4\right )\right )}{x^{11/2}}+\frac {231 \sqrt {2} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {231 \sqrt {2} (-b c+a d)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}}{462 a^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 205, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4}}-\frac {2 c^{3}}{11 a \,x^{\frac {11}{2}}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{7 a^{2} x^{\frac {7}{2}}}\) | \(205\) |
default | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4}}-\frac {2 c^{3}}{11 a \,x^{\frac {11}{2}}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{7 a^{2} x^{\frac {7}{2}}}\) | \(205\) |
risch | \(-\frac {2 \left (231 a^{2} d^{2} x^{4}-231 a b c d \,x^{4}+77 b^{2} c^{2} x^{4}+99 a^{2} c d \,x^{2}-33 a b \,c^{2} x^{2}+21 a^{2} c^{2}\right ) c}{231 a^{3} x^{\frac {11}{2}}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{2 a}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) b c \,d^{2}}{2 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) b^{2} c^{2} d}{2 a^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) b^{3} c^{3}}{2 a^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{3}}{4 a}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) b c \,d^{2}}{4 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) b^{2} c^{2} d}{4 a^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) b^{3} c^{3}}{4 a^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{2 a}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) b c \,d^{2}}{2 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) b^{2} c^{2} d}{2 a^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) b^{3} c^{3}}{2 a^{4}}\) | \(654\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 389, normalized size = 1.28 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a^{3}} - \frac {2 \, {\left (21 \, a^{2} c^{3} + 77 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 33 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1866 vs.
\(2 (226) = 452\).
time = 0.76, size = 1866, normalized size = 6.12 \begin {gather*} \frac {924 \, a^{3} x^{6} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{8} \sqrt {-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}} + {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} x} a^{11} b \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {3}{4}} + {\left (a^{11} b^{4} c^{3} - 3 \, a^{12} b^{3} c^{2} d + 3 \, a^{13} b^{2} c d^{2} - a^{14} b d^{3}\right )} \sqrt {x} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {3}{4}}}{b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}\right ) + 231 \, a^{3} x^{6} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {1}{4}} \log \left (a^{4} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {1}{4}} - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}\right ) - 231 \, a^{3} x^{6} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {1}{4}} \log \left (-a^{4} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{15} b}\right )^{\frac {1}{4}} - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}\right ) - 4 \, {\left (21 \, a^{2} c^{3} + 77 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 33 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {x}}{462 \, a^{3} x^{6}} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 483 vs.
\(2 (226) = 452\).
time = 1.11, size = 483, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} - \frac {2 \, {\left (77 \, b^{2} c^{3} x^{4} - 231 \, a b c^{2} d x^{4} + 231 \, a^{2} c d^{2} x^{4} - 33 \, a b c^{3} x^{2} + 99 \, a^{2} c^{2} d x^{2} + 21 \, a^{2} c^{3}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 1580, normalized size = 5.18 \begin {gather*} -\frac {\frac {2\,c^3}{11\,a}+\frac {2\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{7\,a^2}+\frac {2\,c\,x^4\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a^3}}{x^{11/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{15/4}\,b^{1/4}}+\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{15/4}\,b^{1/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{15/4}\,b^{1/4}}-\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{15/4}\,b^{1/4}}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{15/4}\,b^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{15/4}\,b^{1/4}}+\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{15/4}\,b^{1/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{15/4}\,b^{1/4}}-\frac {\left (\frac {\sqrt {x}\,\left (16\,a^{15}\,b^3\,d^6-96\,a^{14}\,b^4\,c\,d^5+240\,a^{13}\,b^5\,c^2\,d^4-320\,a^{12}\,b^6\,c^3\,d^3+240\,a^{11}\,b^7\,c^4\,d^2-96\,a^{10}\,b^8\,c^5\,d+16\,a^9\,b^9\,c^6\right )}{2}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (-16\,a^{16}\,b^3\,d^3+48\,a^{15}\,b^4\,c\,d^2-48\,a^{14}\,b^5\,c^2\,d+16\,a^{13}\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{15/4}\,b^{1/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{15/4}\,b^{1/4}}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{15/4}\,b^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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