Optimal. Leaf size=448 \[ \frac {\sqrt {a^2 x^2-b} \left (\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}\right )^{3/4} \left (55 a^4 d x^3-41 a^2 b c x^3-87 a^2 b d x+9 b^2 c x\right )}{96 a^2 b^{13/8} \left (a x-\sqrt {b}\right )^2 \left (a x+\sqrt {b}\right )^2}+\frac {\left (41 b c-55 a^2 d\right ) \tan ^{-1}\left (\sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}+\frac {\left (55 a^2 d-41 b c\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {\sqrt [4]{-1} \left (55 a^2 d-41 b c\right ) \tanh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {(-1)^{3/4} \left (55 a^2 d-41 b c\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}+\frac {\left (\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}\right )^{3/4} \left (-55 a^4 d x^2+41 a^2 b c x^2+43 a^2 b d-53 b^2 c\right )}{96 a^3 b^{13/8} \left (a x-\sqrt {b}\right ) \left (a x+\sqrt {b}\right )} \]
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Rubi [B] time = 1.87, antiderivative size = 1077, normalized size of antiderivative = 2.40, number of steps used = 38, number of rules used = 18, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6742, 2122, 288, 290, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 203, 206, 2120, 463, 457} \begin {gather*} -\frac {11 c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {8 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {41 c \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}+\frac {41 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a^3 b^{5/8}}+\frac {55 d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a^3 b^{5/8}}+\frac {55 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a b^{13/8}}+\frac {41 c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {55 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a b^{13/8}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 204
Rule 206
Rule 288
Rule 290
Rule 297
Rule 298
Rule 300
Rule 329
Rule 457
Rule 463
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2120
Rule 2122
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx &=\int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}}+\frac {c x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}}\right ) \, dx\\ &=c \int \frac {x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {(4 c) \operatorname {Subst}\left (\int \frac {x^{7/4} \left (b+x^2\right )^2}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^3}+\frac {(16 d) \operatorname {Subst}\left (\int \frac {x^{15/4}}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {x^{7/4} \left (5 b^2+6 b x^2\right )}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a^3 b}+\frac {(22 d) \operatorname {Subst}\left (\int \frac {x^{7/4}}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {(41 c) \operatorname {Subst}\left (\int \frac {x^{7/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{24 a^3}+\frac {(11 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(41 c) \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 )}{64 a^3}-\frac {(55 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 )}{64 a b}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(41 c) \operatorname {Subst}\left (\int \frac {x^2}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a^3}-\frac {(55 d) \operatorname {Subst}\left (\int \frac {x^2}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a b}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(41 c) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^3 \sqrt {b}}-\frac {(41 c) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^3 \sqrt {b}}+\frac {(55 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 )}{32 a b^{3/2}}+\frac {(55 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 )}{32 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(41 c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^3 \sqrt {b}}+\frac {(41 c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^3 \sqrt {b}}+\frac {(41 c) \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 )}{64 a^3 \sqrt {b}}-\frac {(41 c) \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 )}{64 a^3 \sqrt {b}}+\frac {(55 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 )}{64 a b^{3/2}}-\frac {(55 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 )}{64 a b^{3/2}}-\frac {(55 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 )}{64 a b^{3/2}}+\frac {(55 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 )}{64 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {(41 c) \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 )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {(41 c) \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 )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {(41 c) \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 )}{128 a^3 \sqrt {b}}-\frac {(41 c) \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 )}{128 a^3 \sqrt {b}}+\frac {(55 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 )}{128 \sqrt {2} a b^{13/8}}+\frac {(55 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 )}{128 \sqrt {2} a b^{13/8}}+\frac {(55 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 )}{128 a b^{3/2}}+\frac {(55 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 )}{128 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \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 )}{128 \sqrt {2} a^3 b^{5/8}}+\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}+\frac {41 c \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 )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}-\frac {(41 c) \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 )}{64 \sqrt {2} a^3 b^{5/8}}+\frac {(41 c) \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 )}{64 \sqrt {2} a^3 b^{5/8}}+\frac {(55 d) \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 )}{64 \sqrt {2} a b^{13/8}}-\frac {(55 d) \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 )}{64 \sqrt {2} a b^{13/8}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}+\frac {41 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^3 b^{5/8}}+\frac {55 d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \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 )}{128 \sqrt {2} a^3 b^{5/8}}+\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}+\frac {41 c \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 )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}\\ \end {align*}
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Mathematica [F] time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.38, size = 510, normalized size = 1.14 \begin {gather*} \frac {\left (-53 b^2 c+43 a^2 b d+41 a^2 b c x^2-55 a^4 d x^2\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^3 b^{13/8} \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {\sqrt {-b+a^2 x^2} \left (9 b^2 c x-87 a^2 b d x-41 a^2 b c x^3+55 a^4 d x^3\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^2 b^{13/8} \left (-\sqrt {b}+a x\right )^2 \left (\sqrt {b}+a x\right )^2}+\frac {\left (41 b c-55 a^2 d\right ) \tan ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}+\frac {\left (-41 b c+55 a^2 d\right ) \tan ^{-1}\left (\frac {-1+\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{64 \sqrt {2} a^3 b^{13/8}}+\frac {\left (-41 b c+55 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {\left (-41 b c+55 a^2 d\right ) \tanh ^{-1}\left (\frac {1+\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{64 \sqrt {2} a^3 b^{13/8}} \end {gather*}
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.05, size = 4991, normalized size = 11.14
result too large to display
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 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}}}{\left (a^{2} x^{2}-b \right )^{\frac {5}{2}}}\, dx\]
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 {{\left (c x^{2} + d\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {5}{2}}}\,{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 {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{3/4}\,\left (c\,x^2+d\right )}{{\left (a^2\,x^2-b\right )}^{5/2}} \,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 {\left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {3}{4}} \left (c x^{2} + d\right )}{\left (a^{2} x^{2} - b\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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