Optimal. Leaf size=288 \[ \frac {\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}+\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{13/4}}+\frac {2 \sqrt {x} (b c-a d)^2}{d^3}-\frac {2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d} \]
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Rubi [A] time = 0.24, antiderivative size = 288, 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} \[ -\frac {2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac {2 \sqrt {x} (b c-a d)^2}{d^3}+\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}+\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{13/4}}+\frac {2 b^2 x^{9/2}}{9 d} \]
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 (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {b (b c-2 a d) x^{3/2}}{d^2}+\frac {b^2 x^{7/2}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^{3/2}}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}+\frac {(b c-a d)^2 \int \frac {x^{3/2}}{c+d x^2} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}-\frac {\left (c (b c-a d)^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{d^3}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}-\frac {\left (2 c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}-\frac {\left (\sqrt {c} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (\sqrt {c} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}-\frac {\left (\sqrt {c} (b c-a d)^2\right ) \operatorname {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 )}{2 d^{7/2}}-\frac {\left (\sqrt {c} (b c-a d)^2\right ) \operatorname {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 )}{2 d^{7/2}}+\frac {\left (\sqrt [4]{c} (b c-a d)^2\right ) \operatorname {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 )}{2 \sqrt {2} d^{13/4}}+\frac {\left (\sqrt [4]{c} (b c-a d)^2\right ) \operatorname {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 )}{2 \sqrt {2} d^{13/4}}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}+\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\left (\sqrt [4]{c} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}+\frac {\left (\sqrt [4]{c} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}\\ &=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}+\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}+\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 276, normalized size = 0.96 \[ \frac {-72 b d^{5/4} x^{5/2} (b c-2 a d)+360 \sqrt [4]{d} \sqrt {x} (b c-a d)^2+45 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-45 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+90 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )-90 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )+40 b^2 d^{9/4} x^{9/2}}{180 d^{13/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 1268, normalized size = 4.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 385, normalized size = 1.34 \[ -\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{4}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{4}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{4}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{8} x^{\frac {9}{2}} - 9 \, b^{2} c d^{7} x^{\frac {5}{2}} + 18 \, a b d^{8} x^{\frac {5}{2}} + 45 \, b^{2} c^{2} d^{6} \sqrt {x} - 90 \, a b c d^{7} \sqrt {x} + 45 \, a^{2} d^{8} \sqrt {x}\right )}}{45 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 495, normalized size = 1.72 \[ \frac {2 b^{2} x^{\frac {9}{2}}}{9 d}+\frac {4 a b \,x^{\frac {5}{2}}}{5 d}-\frac {2 b^{2} c \,x^{\frac {5}{2}}}{5 d^{2}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 d}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 d}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{d^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{d^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{2 d^{2}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 d^{3}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 d^{3}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 d^{3}}+\frac {2 a^{2} \sqrt {x}}{d}-\frac {4 a b c \sqrt {x}}{d^{2}}+\frac {2 b^{2} c^{2} \sqrt {x}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 324, normalized size = 1.12 \[ -\frac {{\left (\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{4 \, d^{3}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 9 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 1175, normalized size = 4.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 53.53, size = 656, normalized size = 2.28 \[ \begin {cases} \tilde {\infty } \left (2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}}{c} & \text {for}\: d = 0 \\\frac {2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}}{d} & \text {for}\: c = 0 \\\frac {\sqrt [4]{-1} a^{2} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{2 d} - \frac {\sqrt [4]{-1} a^{2} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} \log {\left (\sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{2 d} + \frac {\sqrt [4]{-1} a^{2} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{c} \sqrt [4]{\frac {1}{d}}} \right )}}{d} + \frac {2 a^{2} \sqrt {x}}{d} - \frac {\sqrt [4]{-1} a b c^{\frac {5}{4}} \sqrt [4]{\frac {1}{d}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{d^{2}} + \frac {\sqrt [4]{-1} a b c^{\frac {5}{4}} \sqrt [4]{\frac {1}{d}} \log {\left (\sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{d^{2}} - \frac {2 \sqrt [4]{-1} a b c^{\frac {5}{4}} \sqrt [4]{\frac {1}{d}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{c} \sqrt [4]{\frac {1}{d}}} \right )}}{d^{2}} - \frac {4 a b c \sqrt {x}}{d^{2}} + \frac {4 a b x^{\frac {5}{2}}}{5 d} + \frac {\sqrt [4]{-1} b^{2} c^{\frac {9}{4}} \sqrt [4]{\frac {1}{d}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{2 d^{3}} - \frac {\sqrt [4]{-1} b^{2} c^{\frac {9}{4}} \sqrt [4]{\frac {1}{d}} \log {\left (\sqrt [4]{-1} \sqrt [4]{c} \sqrt [4]{\frac {1}{d}} + \sqrt {x} \right )}}{2 d^{3}} + \frac {\sqrt [4]{-1} b^{2} c^{\frac {9}{4}} \sqrt [4]{\frac {1}{d}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{c} \sqrt [4]{\frac {1}{d}}} \right )}}{d^{3}} + \frac {2 b^{2} c^{2} \sqrt {x}}{d^{3}} - \frac {2 b^{2} c x^{\frac {5}{2}}}{5 d^{2}} + \frac {2 b^{2} x^{\frac {9}{2}}}{9 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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