Optimal. Leaf size=367 \[ \frac {(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 \sqrt {x} (b c-5 a d)}{2 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 367, 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 = {466, 468, 570, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 \sqrt {x} (b c-5 a d)}{2 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 466
Rule 468
Rule 570
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^4 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (7 b c-3 a d)+d (b c-5 a d) x^4\right )}{x^4 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {d^2 (b c-5 a d)}{b}+\frac {c^2 (-7 b c+3 a d)}{a x^4}+\frac {(-b c+a d)^2 (7 b c+5 a d)}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2} b^2}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2} b^2}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{8 a^{5/2} b^{5/2}}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{8 a^{5/2} b^{5/2}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{8 \sqrt {2} a^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{8 \sqrt {2} a^{11/4} b^{9/4}}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 )}{4 \sqrt {2} a^{11/4} b^{9/4}}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}\\ \end {align*}
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Mathematica [C] time = 2.67, size = 350, normalized size = 0.95 \begin {gather*} -\frac {32768 b^3 x^6 \left (c+d x^2\right )^3 \, _5F_4\left (\frac {1}{4},2,2,2,2;1,1,1,\frac {17}{4};-\frac {b x^2}{a}\right )+39 a \left (15 a^2 \left (2187 c^3+6561 c^2 d x^2+6561 c d^2 x^4+1547 d^3 x^6\right )+6 a b x^2 \left (469 c^3+1407 c^2 d x^2+2367 c d^2 x^4+789 d^3 x^6\right )-5 b^2 x^4 \left (-869 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right )\right )-585 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right ) \left (a^3 \left (2187 c^3+6561 c^2 d x^2+6561 c d^2 x^4+1547 d^3 x^6\right )+a^2 b x^2 \left (625 c^3+1875 c^2 d x^2+2259 c d^2 x^4+625 d^3 x^6\right )+a b^2 x^4 \left (c^3+1155 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right )+b^3 x^6 \left (-869 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right )\right )}{149760 a^3 b^2 x^{11/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.55, size = 244, normalized size = 0.66 \begin {gather*} \frac {(5 a d+7 b c) (a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(5 a d+7 b c) (a d-b c)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {15 a^3 d^3 x^2-9 a^2 b c d^2 x^2+12 a^2 b d^3 x^4-4 a b^2 c^3+9 a b^2 c^2 d x^2-7 b^3 c^3 x^2}{6 a^2 b^2 x^{3/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.42, size = 1967, normalized size = 5.36
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 501, normalized size = 1.37 \begin {gather*} \frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {2 \, c^{3}}{3 \, a^{2} x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \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} + 5 \, \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 )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \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} + 5 \, \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 )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \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} + 5 \, \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 )}{16 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \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} + 5 \, \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 )}{16 \, a^{3} b^{3}} - \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 682, normalized size = 1.86 \begin {gather*} \frac {a \,d^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {3 c^{2} d \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a}-\frac {b \,c^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 c \,d^{2} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {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 )}{16 a b}+\frac {9 \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 )}{8 a^{2}}+\frac {9 \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 )}{8 a^{2}}+\frac {9 \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 )}{16 a^{2}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,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 )}{16 a^{3}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{2}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{2}}-\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {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 )}{16 b^{2}}+\frac {2 d^{3} \sqrt {x}}{b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 415, normalized size = 1.13 \begin {gather*} \frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{2}}{6 \, {\left (a^{2} b^{3} x^{\frac {7}{2}} + a^{3} b^{2} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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}}}}{16 \, a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 1759, normalized size = 4.79
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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