Optimal. Leaf size=304 \[ -\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {2 d \sqrt {x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b} \]
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Rubi [A] time = 0.25, antiderivative size = 304, 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 = {461, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {2 d \sqrt {x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 461
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\sqrt {x} \left (a+b x^2\right )} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 \sqrt {x}}+\frac {d^2 (3 b c-a d) x^{3/2}}{b^2}+\frac {d^3 x^{7/2}}{b}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{b^3 \sqrt {x} \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 ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {\left (2 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b^3}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \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 \sqrt {a} b^{7/2}}+\frac {(b c-a d)^3 \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 \sqrt {a} b^{7/2}}-\frac {(b c-a d)^3 \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} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \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} a^{3/4} b^{13/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 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^{3/4} b^{13/4}}+\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^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \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} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \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} a^{3/4} b^{13/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 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^{3/4} b^{13/4}}+\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^{3/4} b^{13/4}}-\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^{3/4} b^{13/4}}+\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^{3/4} b^{13/4}}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 96, normalized size = 0.32 \[ \frac {2 \sqrt {x} \left (a d \left (45 a^2 d^2-9 a b d \left (15 c+d x^2\right )+b^2 \left (135 c^2+27 c d x^2+5 d^2 x^4\right )\right )+45 (b c-a d)^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{45 a b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 1862, normalized size = 6.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 490, normalized size = 1.61 \[ \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{4}} + \frac {2 \, {\left (5 \, b^{8} d^{3} x^{\frac {9}{2}} + 27 \, b^{8} c d^{2} x^{\frac {5}{2}} - 9 \, a b^{7} d^{3} x^{\frac {5}{2}} + 135 \, b^{8} c^{2} d \sqrt {x} - 135 \, a b^{7} c d^{2} \sqrt {x} + 45 \, a^{2} b^{6} d^{3} \sqrt {x}\right )}}{45 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 650, normalized size = 2.14 \[ \frac {2 d^{3} x^{\frac {9}{2}}}{9 b}-\frac {2 a \,d^{3} x^{\frac {5}{2}}}{5 b^{2}}+\frac {6 c \,d^{2} x^{\frac {5}{2}}}{5 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} 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^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a c \,d^{2} \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 \,d^{2} \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 \,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^{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 a}+\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 a}+\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 a}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, 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}+\frac {2 a^{2} d^{3} \sqrt {x}}{b^{3}}-\frac {6 a c \,d^{2} \sqrt {x}}{b^{2}}+\frac {6 c^{2} d \sqrt {x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.44, size = 390, normalized size = 1.28 \[ \frac {2 \, {\left (5 \, b^{2} d^{3} x^{\frac {9}{2}} + 9 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {x}\right )}}{45 \, b^{3}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 1460, normalized size = 4.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 71.75, size = 874, normalized size = 2.88 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 c^{3} \sqrt {x} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 d^{3} x^{\frac {13}{2}}}{13}}{a} & \text {for}\: b = 0 \\\frac {- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}}{b} & \text {for}\: a = 0 \\\frac {\sqrt [4]{-1} a^{\frac {9}{4}} d^{3} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{3}} - \frac {\sqrt [4]{-1} a^{\frac {9}{4}} d^{3} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{3}} + \frac {\sqrt [4]{-1} a^{\frac {9}{4}} d^{3} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{3}} - \frac {3 \sqrt [4]{-1} a^{\frac {5}{4}} c d^{2} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} + \frac {3 \sqrt [4]{-1} a^{\frac {5}{4}} c d^{2} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} - \frac {3 \sqrt [4]{-1} a^{\frac {5}{4}} c d^{2} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{2}} + \frac {3 \sqrt [4]{-1} \sqrt [4]{a} c^{2} d \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b} - \frac {3 \sqrt [4]{-1} \sqrt [4]{a} c^{2} d \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b} + \frac {3 \sqrt [4]{-1} \sqrt [4]{a} c^{2} d \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b} + \frac {2 a^{2} d^{3} \sqrt {x}}{b^{3}} - \frac {6 a c d^{2} \sqrt {x}}{b^{2}} - \frac {2 a d^{3} x^{\frac {5}{2}}}{5 b^{2}} + \frac {6 c^{2} d \sqrt {x}}{b} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5 b} + \frac {2 d^{3} x^{\frac {9}{2}}}{9 b} - \frac {\sqrt [4]{-1} c^{3} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} + \frac {\sqrt [4]{-1} c^{3} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {3}{4}}} - \frac {\sqrt [4]{-1} c^{3} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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