Optimal. Leaf size=376 \[ -\frac {(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.42, antiderivative size = 376, 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 {c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}-\frac {(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \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^{9/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^8 \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^{7/2} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (11 b c-7 a d)-d (3 b c+a d) x^4\right )}{x^8 \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^{7/2} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {c^2 (-11 b c+7 a d)}{a x^8}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{a^2 x^4}-\frac {(-b c+a d)^2 (11 b c+a d)}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+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^{7/2} b}+\frac {\left ((b c-a d)^2 (11 b c+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^{7/2} b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+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^{7/2} b^{3/2}}+\frac {\left ((b c-a d)^2 (11 b c+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^{7/2} b^{3/2}}-\frac {\left ((b c-a d)^2 (11 b c+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^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+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^{15/4} b^{5/4}}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left ((b c-a d)^2 (11 b c+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^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+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^{15/4} b^{5/4}}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}\\ \end {align*}
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Mathematica [C] time = 2.40, size = 353, normalized size = 0.94 \begin {gather*} -\frac {-229376 a b^2 x^4 \left (c+d x^2\right )^3 \, _5F_4\left (-\frac {3}{4},2,2,2,2;1,1,1,\frac {13}{4};-\frac {b x^2}{a}\right )-15 a \left (21 a^2 \left (625 c^3+1875 c^2 d x^2+1875 c d^2 x^4+241 d^3 x^6\right )+6 a b x^2 \left (-1195 c^3-6657 c^2 d x^2+2751 c d^2 x^4+917 d^3 x^6\right )-7 b^2 x^4 \left (-1823 c^3+7203 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right )\right )+315 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right ) \left (a^3 \left (625 c^3+1875 c^2 d x^2+1875 c d^2 x^4+241 d^3 x^6\right )+3 a^2 b x^2 \left (c^3+3 c^2 d x^2-1149 c d^2 x^4+d^3 x^6\right )+9 a b^2 x^4 \left (27 c^3+977 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right )+b^3 x^6 \left (-1823 c^3+7203 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right )\right )}{241920 a^4 b x^{11/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.64, size = 255, normalized size = 0.68 \begin {gather*} -\frac {(a d+11 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^{15/4} b^{5/4}}+\frac {(a d+11 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^{15/4} b^{5/4}}+\frac {-21 a^3 d^3 x^4-12 a^2 b c^3-84 a^2 b c^2 d x^2+63 a^2 b c d^2 x^4+44 a b^2 c^3 x^2-147 a b^2 c^2 d x^4+77 b^3 c^3 x^4}{42 a^3 b x^{7/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 1955, normalized size = 5.20
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 = 509, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \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 )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \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 )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \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 )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \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 )}{16 \, a^{4} b^{2}} + \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^{3} b} + \frac {2 \, {\left (14 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{3} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 706, normalized size = 1.88 \begin {gather*} \frac {3 c \,d^{2} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a}-\frac {3 b \,c^{2} d \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {b^{2} c^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a^{3}}-\frac {d^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b}+\frac {\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 a b}+\frac {\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 a b}+\frac {\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 a b}+\frac {9 \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^{2}}+\frac {9 \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^{2}}+\frac {9 \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^{2}}-\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{3}}-\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{3}}-\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \,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^{3}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{4}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{4}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{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 )}{16 a^{4}}-\frac {2 c^{2} d}{a^{2} x^{\frac {3}{2}}}+\frac {4 b \,c^{3}}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 c^{3}}{7 a^{2} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.55, size = 424, normalized size = 1.13 \begin {gather*} -\frac {12 \, a^{2} b c^{3} - 7 \, {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{4} - 4 \, {\left (11 \, a b^{2} c^{3} - 21 \, a^{2} b c^{2} d\right )} x^{2}}{42 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + a^{4} b x^{\frac {7}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, 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 (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, 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 (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, 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 (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, 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}}}}{16 \, a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 1746, normalized size = 4.64
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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