Optimal. Leaf size=306 \[ \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}} \]
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Rubi [A]
time = 0.18, antiderivative size = 306, 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 = {472, 335, 303,
1176, 631, 210, 1179, 642} \begin {gather*} \frac {2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\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} \sqrt [4]{a} b^{15/4}}-\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} \sqrt [4]{a} b^{15/4}}+\frac {2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 472
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \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 ) \sqrt {x}}{b^3}+\frac {d^2 (3 b c-a d) x^{5/2}}{b^2}+\frac {d^3 x^{9/2}}{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 ) \sqrt {x}}{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^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^2}{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 ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\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 )}{b^{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 )}{b^{7/2}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 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 b^4}+\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 b^4}+\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} \sqrt [4]{a} b^{15/4}}+\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} \sqrt [4]{a} b^{15/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 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} \sqrt [4]{a} b^{15/4}}-\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} \sqrt [4]{a} b^{15/4}}+\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} \sqrt [4]{a} b^{15/4}}-\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} \sqrt [4]{a} b^{15/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 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} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\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} \sqrt [4]{a} b^{15/4}}-\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} \sqrt [4]{a} b^{15/4}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 188, normalized size = 0.61 \begin {gather*} \frac {2 d x^{3/2} \left (77 a^2 d^2-33 a b d \left (7 c+d x^2\right )+3 b^2 \left (77 c^2+33 c d x^2+7 d^2 x^4\right )\right )}{231 b^3}+\frac {(-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 {2} \sqrt [4]{a} b^{15/4}}+\frac {(-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 {2} \sqrt [4]{a} b^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 208, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(208\) |
default | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(208\) |
risch | \(\frac {2 \left (21 b^{2} d^{2} x^{4}-33 a b \,d^{2} x^{2}+99 b^{2} c d \,x^{2}+77 a^{2} d^{2}-231 a b c d +231 b^{2} c^{2}\right ) d \,x^{\frac {3}{2}}}{231 b^{3}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) a^{3} d^{3}}{2 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) a^{2} c \,d^{2}}{2 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) a \,c^{2} d}{2 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3}}{2 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) a^{3} d^{3}}{2 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) a^{2} c \,d^{2}}{2 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) a \,c^{2} d}{2 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3}}{2 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\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 ) a^{3} d^{3}}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \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 ) a^{2} c \,d^{2}}{4 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {3 \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 ) a \,c^{2} d}{4 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\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 ) c^{3}}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(648\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 282, normalized size = 0.92 \begin {gather*} \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 )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{3}} + \frac {2 \, {\left (21 \, b^{2} d^{3} x^{\frac {11}{2}} + 33 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{231 \, b^{3}} \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. 2441 vs.
\(2 (227) = 454\).
time = 0.65, size = 2441, normalized size = 7.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs.
\(2 (291) = 582\).
time = 18.28, size = 649, normalized size = 2.12 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}}{b} & \text {for}\: a = 0 \\\frac {\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}}{a} & \text {for}\: b = 0 \\\frac {2 a^{2} d^{3} x^{\frac {3}{2}}}{3 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} - \frac {2 a c d^{2} x^{\frac {3}{2}}}{b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {2 a d^{3} x^{\frac {7}{2}}}{7 b^{2}} + \frac {2 c^{2} d x^{\frac {3}{2}}}{b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7 b} + \frac {2 d^{3} x^{\frac {11}{2}}}{11 b} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} & \text {otherwise} \end {cases} \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. 490 vs.
\(2 (227) = 454\).
time = 1.61, size = 490, normalized size = 1.60 \begin {gather*} \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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^{6}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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^{6}} + \frac {2 \, {\left (21 \, b^{10} d^{3} x^{\frac {11}{2}} + 99 \, b^{10} c d^{2} x^{\frac {7}{2}} - 33 \, a b^{9} d^{3} x^{\frac {7}{2}} + 231 \, b^{10} c^{2} d x^{\frac {3}{2}} - 231 \, a b^{9} c d^{2} x^{\frac {3}{2}} + 77 \, a^{2} b^{8} d^{3} x^{\frac {3}{2}}\right )}}{231 \, b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 574, normalized size = 1.88 \begin {gather*} x^{3/2}\,\left (\frac {2\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{3\,b}\right )-x^{7/2}\,\left (\frac {2\,a\,d^3}{7\,b^2}-\frac {6\,c\,d^2}{7\,b}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,b^{15/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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