Optimal. Leaf size=876 \[ \frac {\left (-97 b^2 c-a^2 b d+45 a^2 b c x^2-51 a^4 d x^2\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{96 a^2 b^{15/8} \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {\sqrt {-b+a^2 x^2} \left (13 b^2 c x-83 a^2 b d x-45 a^2 b c x^3+51 a^4 d x^3\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{96 a b^{15/8} \left (-\sqrt {b}+a x\right )^2 \left (\sqrt {b}+a x\right )^2}+\frac {5 \left (-3 b c+29 a^2 d\right ) \text {ArcTan}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} d \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}+\frac {5 \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {5 (-1)^{3/4} \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {5 \sqrt [4]{-1} \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.29, antiderivative size = 1407, normalized size of antiderivative = 1.61, number of steps
used = 52, number of rules used = 20, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.408, Rules used = {6874, 2145,
477, 481, 592, 541, 536, 220, 218, 212, 209, 217, 1179, 642, 1176, 631, 210, 468, 294, 335}
\begin {gather*} \frac {8 c \left (a x+\sqrt {a^2 x^2-b}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}-\frac {5 c \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {8 d \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {15 c \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 b \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {7 d \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}-\frac {2 d \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {\sqrt {2} d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {\sqrt {2} d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{(-b)^{15/8}}+\frac {15 c \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}-\frac {15 c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {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 )}{\sqrt {2} (-b)^{15/8}}-\frac {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 )}{\sqrt {2} (-b)^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {145 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} b^{15/8}}-\frac {15 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^2 b^{7/8}}+\frac {145 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} b^{15/8}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 220
Rule 294
Rule 335
Rule 468
Rule 477
Rule 481
Rule 536
Rule 541
Rule 592
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2145
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx &=\int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}}+\frac {c x \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{5/2}}\right ) \, dx\\ &=c \int \frac {x \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {(8 c) \text {Subst}\left (\int \frac {x^{13/4} \left (b+x^2\right )}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^2}+(32 d) \text {Subst}\left (\int \frac {x^{21/4}}{\left (-b+x^2\right )^4 \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {(10 c) \text {Subst}\left (\int \frac {x^{13/4}}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a^2}+(128 d) \text {Subst}\left (\int \frac {x^{24}}{\left (-b+x^8\right )^4 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {(15 c) \text {Subst}\left (\int \frac {x^{5/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^2}-\frac {(8 d) \text {Subst}\left (\int \frac {x^8 \left (-9 b^2-33 b x^8\right )}{\left (-b+x^8\right )^3 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{3 b}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(15 c) \text {Subst}\left (\int \frac {1}{x^{3/4} \left (-b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{64 a^2}-\frac {d \text {Subst}\left (\int \frac {-42 b^3-426 b^2 x^8}{\left (-b+x^8\right )^2 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{12 b^2}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(15 c) \text {Subst}\left (\int \frac {1}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a^2}-\frac {d \text {Subst}\left (\int \frac {204 b^4+3276 b^3 x^8}{\left (-b+x^8\right ) \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{192 b^4}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^2 \sqrt {b}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^2 \sqrt {b}}-\frac {(8 d) \text {Subst}\left (\int \frac {1}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}-\frac {(145 d) \text {Subst}\left (\int \frac {1}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 b}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(4 d) \text {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{3/2}}-\frac {(4 d) \text {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{3/2}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 b^{3/2}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 b^{3/2}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {(15 c) \text {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^2 b^{7/8}}+\frac {(15 c) \text {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^2 b^{7/8}}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(15 c) \text {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^2 b^{3/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(2 d) \text {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 )}{(-b)^{7/4}}-\frac {(2 d) \text {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 )}{(-b)^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}+\frac {(145 d) \text {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 b^{7/4}}+\frac {(145 d) \text {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 b^{7/4}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {15 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^2 b^{7/8}}-\frac {(15 c) \text {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^2 b^{7/8}}+\frac {(15 c) \text {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^2 b^{7/8}}+\frac {d \text {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)^{15/8}}+\frac {d \text {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)^{15/8}}-\frac {d \text {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 )}{(-b)^{7/4}}-\frac {d \text {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 )}{(-b)^{7/4}}-\frac {(145 d) \text {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} b^{15/8}}-\frac {(145 d) \text {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} b^{15/8}}+\frac {(145 d) \text {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 b^{7/4}}+\frac {(145 d) \text {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 b^{7/4}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {15 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^2 b^{7/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\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)^{15/8}}-\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)^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {145 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} b^{15/8}}-\frac {15 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^2 b^{7/8}}+\frac {145 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} b^{15/8}}-\frac {\left (\sqrt {2} d\right ) \text {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)^{15/8}}+\frac {\left (\sqrt {2} d\right ) \text {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)^{15/8}}+\frac {(145 d) \text {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} b^{15/8}}-\frac {(145 d) \text {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} b^{15/8}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\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)^{15/8}}-\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)^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {145 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} b^{15/8}}-\frac {15 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^2 b^{7/8}}+\frac {145 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} b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\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)^{15/8}}-\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)^{15/8}}+\frac {15 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^2 b^{7/8}}-\frac {145 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} b^{15/8}}-\frac {15 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^2 b^{7/8}}+\frac {145 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} b^{15/8}}\\ \end {align*}
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Mathematica [F]
time = 10.32, 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 )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}{x \left (a^{2} x^{2}-b \right )^{\frac {5}{2}}}\, dx\]
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*} \text {Failed to integrate} \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. 5382 vs.
\(2 (693) = 1386\).
time = 13.76, size = 5382, normalized size = 6.14 \begin {gather*} \text {Too large to display} \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 {5}{4}} \left (c x^{2} + d\right )}{x \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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Giac [F(-1)] Timed out
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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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 )}^{5/4}\,\left (c\,x^2+d\right )}{x\,{\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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