Optimal. Leaf size=326 \[ \frac {2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{17/4}}+\frac {2 \sqrt {x} (b c-a d)^3}{b^4}+\frac {2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b} \]
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Rubi [A] time = 0.28, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {461, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac {2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac {2 \sqrt {x} (b c-a d)^3}{b^4}+\frac {\sqrt [4]{a} (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} b^{17/4}}-\frac {\sqrt [4]{a} (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} b^{17/4}}+\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{17/4}}+\frac {2 d^3 x^{13/2}}{13 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 329
Rule 461
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{b^3}+\frac {d^2 (3 b c-a d) x^{7/2}}{b^2}+\frac {d^3 x^{11/2}}{b}+\frac {\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) x^{3/2}}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}+\frac {(b c-a d)^3 \int \frac {x^{3/2}}{a+b x^2} \, dx}{b^3}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}-\frac {\left (a (b c-a d)^3\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{b^4}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}-\frac {\left (2 a (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}-\frac {\left (\sqrt {a} (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^4}-\frac {\left (\sqrt {a} (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}-\frac {\left (\sqrt {a} (b c-a d)^3\right ) \operatorname {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 b^{9/2}}-\frac {\left (\sqrt {a} (b c-a d)^3\right ) \operatorname {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 b^{9/2}}+\frac {\left (\sqrt [4]{a} (b c-a d)^3\right ) \operatorname {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} b^{17/4}}+\frac {\left (\sqrt [4]{a} (b c-a d)^3\right ) \operatorname {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} b^{17/4}}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}+\frac {\sqrt [4]{a} (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} b^{17/4}}-\frac {\sqrt [4]{a} (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} b^{17/4}}-\frac {\left (\sqrt [4]{a} (b c-a d)^3\right ) \operatorname {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} b^{17/4}}+\frac {\left (\sqrt [4]{a} (b c-a d)^3\right ) \operatorname {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} b^{17/4}}\\ &=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}+\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (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} b^{17/4}}-\frac {\sqrt [4]{a} (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} b^{17/4}}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 133, normalized size = 0.41 \begin {gather*} \frac {2 \sqrt {x} \left (-585 a^3 d^3+117 a^2 b d^2 \left (15 c+d x^2\right )-13 a b^2 d \left (135 c^2+27 c d x^2+5 d^2 x^4\right )-585 (b c-a d)^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right )+3 b^3 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )\right )}{585 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 254, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {x} \left (-585 a^3 d^3+1755 a^2 b c d^2+117 a^2 b d^3 x^2-1755 a b^2 c^2 d-351 a b^2 c d^2 x^2-65 a b^2 d^3 x^4+585 b^3 c^3+351 b^3 c^2 d x^2+195 b^3 c d^2 x^4+45 b^3 d^3 x^6\right )}{585 b^4}-\frac {\sqrt [4]{a} (a d-b c)^3 \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (a d-b c)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.49, size = 1898, normalized size = 5.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 531, normalized size = 1.63 \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 \, b^{5}} - \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 \, b^{5}} - \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 \, b^{5}} + \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 \, b^{5}} + \frac {2 \, {\left (45 \, b^{12} d^{3} x^{\frac {13}{2}} + 195 \, b^{12} c d^{2} x^{\frac {9}{2}} - 65 \, a b^{11} d^{3} x^{\frac {9}{2}} + 351 \, b^{12} c^{2} d x^{\frac {5}{2}} - 351 \, a b^{11} c d^{2} x^{\frac {5}{2}} + 117 \, a^{2} b^{10} d^{3} x^{\frac {5}{2}} + 585 \, b^{12} c^{3} \sqrt {x} - 1755 \, a b^{11} c^{2} d \sqrt {x} + 1755 \, a^{2} b^{10} c d^{2} \sqrt {x} - 585 \, a^{3} b^{9} d^{3} \sqrt {x}\right )}}{585 \, b^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 712, normalized size = 2.18 \begin {gather*} \frac {2 d^{3} x^{\frac {13}{2}}}{13 b}-\frac {2 a \,d^{3} x^{\frac {9}{2}}}{9 b^{2}}+\frac {2 c \,d^{2} x^{\frac {9}{2}}}{3 b}+\frac {2 a^{2} d^{3} x^{\frac {5}{2}}}{5 b^{3}}-\frac {6 a c \,d^{2} x^{\frac {5}{2}}}{5 b^{2}}+\frac {6 c^{2} d \,x^{\frac {5}{2}}}{5 b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 b^{4}}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{3}}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{3}}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} c \,d^{2} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 b^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,c^{2} d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 b^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 b}-\frac {2 a^{3} d^{3} \sqrt {x}}{b^{4}}+\frac {6 a^{2} c \,d^{2} \sqrt {x}}{b^{3}}-\frac {6 a \,c^{2} d \sqrt {x}}{b^{2}}+\frac {2 c^{3} \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 437, normalized size = 1.34 \begin {gather*} -\frac {{\left (\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}}}\right )} a}{4 \, b^{4}} + \frac {2 \, {\left (45 \, b^{3} d^{3} x^{\frac {13}{2}} + 65 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{\frac {9}{2}} + 117 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {5}{2}} + 585 \, {\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 )}}{585 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 1564, normalized size = 4.80
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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