Optimal. Leaf size=535 \[ \frac {\sqrt {2+\sqrt {2}} d \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{5/8}}+\frac {4 c \left (260 a^6 x^6+325 a^4 b x^4-676 a^2 b^2 x^2+128 b^3\right )+4 c \sqrt {a^2 x^2-b} \left (260 a^5 x^5+455 a^3 b x^3-416 a b^2 x\right )}{715 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}} \]
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Rubi [A] time = 2.11, antiderivative size = 490, normalized size of antiderivative = 0.92, number of steps used = 20, number of rules used = 14, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6742, 2120, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 203, 206, 270} \begin {gather*} -\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {\sqrt {2} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {b^3 c}{26 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}+\frac {b c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{2 a^4}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}}{22 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 270
Rule 297
Rule 298
Rule 300
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2120
Rule 6742
Rubi steps
\begin {align*} \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\int \left (\frac {d}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\frac {c x^3}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \, dx\\ &=c \int \frac {x^3}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx+d \int \frac {1}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\\ &=\frac {c \operatorname {Subst}\left (\int \frac {\left (b+x^2\right )^3}{x^{17/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^4}+(2 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=\frac {c \operatorname {Subst}\left (\int \left (\frac {b^3}{x^{17/4}}+\frac {3 b^2}{x^{9/4}}+\frac {3 b}{\sqrt [4]{x}}+x^{7/4}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^4}+(8 d) \operatorname {Subst}\left (\int \frac {x^2}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}-\frac {(4 d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {(4 d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}\\ &=-\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b}-x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b}+x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}\\ &=-\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}+2 x}{-\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {d \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}-2 x}{-\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}\\ &=-\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {\left (\sqrt {2} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\left (\sqrt {2} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}\\ &=-\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {\sqrt {2} d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}\\ \end {align*}
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Mathematica [C] time = 5.18, size = 742, normalized size = 1.39 \begin {gather*} \frac {4 \left (\frac {13585 a^5 d \sqrt {\text {sgn}(a)^2} \left (\sqrt {a^2 x^2-b}+a x\right )}{\sqrt {a^2} \text {sgn}(a)}+\frac {13585 a^4 d \sqrt {a^2 x^2-b} \left (\sqrt {a^2 x^2-b}+a x\right )^2 \left (2 \, _2F_1\left (\frac {3}{8},1;\frac {11}{8};-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^2}{b}\right )-1\right )}{a x \left (\sqrt {a^2 x^2-b}+a x\right )-b}-\frac {3 c \sqrt {a^2 x^2-b} \left (156 a b^3 x \left (8 \sqrt {a^2 x^2-b}+13 a x\right )+5720 a^7 x^7 \left (\sqrt {a^2 x^2-b}+a x\right )-260 a^5 b x^5 \left (14 \sqrt {a^2 x^2-b}+25 a x\right )-65 a^3 b^2 x^3 \left (21 \sqrt {a^2 x^2-b}+4 a x\right )-384 b^4\right )}{\left (\sqrt {a^2 x^2-b}+a x\right )^2 \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )}+\frac {3 c \sqrt {a^2 x^2-b} \left (b-2 a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^4 \left (-832 a b^3 x \left (8 \sqrt {a^2 x^2-b}+13 a x\right )+5720 a^7 x^7 \left (\sqrt {a^2 x^2-b}+a x\right )+260 a^5 b x^5 \left (5 \sqrt {a^2 x^2-b}-6 a x\right )+455 a^3 b^2 x^3 \left (16 \sqrt {a^2 x^2-b}+13 a x\right )+2048 b^4\right )}{-a b^5 x \left (11 \sqrt {a^2 x^2-b}+61 a x\right )+1024 a^{11} x^{11} \left (\sqrt {a^2 x^2-b}+a x\right )-256 a^9 b x^9 \left (11 \sqrt {a^2 x^2-b}+13 a x\right )+256 a^7 b^2 x^7 \left (11 \sqrt {a^2 x^2-b}+16 a x\right )-112 a^5 b^3 x^5 \left (11 \sqrt {a^2 x^2-b}+21 a x\right )+20 a^3 b^4 x^3 \left (11 \sqrt {a^2 x^2-b}+31 a x\right )+b^6}\right )}{40755 a^4 b \sqrt [4]{\sqrt {a^2 x^2-b}+a x}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.95, size = 510, normalized size = 0.95 \begin {gather*} \frac {4 c \sqrt {-b+a^2 x^2} \left (-416 a b^2 x+455 a^3 b x^3+260 a^5 x^5\right )+4 c \left (128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6\right )}{715 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {\sqrt {2+\sqrt {2}} d \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 842, normalized size = 1.57 \begin {gather*} -\frac {2860 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (-\frac {d^{8} + \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} - \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} - \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\right ) + 2860 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (\frac {d^{8} - \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} + \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\right ) - 715 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + 4 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + 4 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}\right ) + 715 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} - 4 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + 4 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}\right ) - 5720 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (-\frac {\left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} - \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\right ) + 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + 8 \, {\left (55 \, a^{4} c x^{4} + 36 \, a^{2} b c x^{2} - 128 \, b^{2} c - {\left (55 \, a^{3} c x^{3} + 96 \, a b c x\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{1430 \, a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{4}+d}{x \sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c\,x^4+d}{x\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{x \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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