Integrand size = 46, antiderivative size = 384 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-b+a^2 x^2} \left (-9 b^{9/8} c-5 a^2 \sqrt [8]{b} d+4 a^2 \sqrt [8]{b} c x^2\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^3 \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {4 \sqrt [8]{b} c x \sqrt [4]{\frac {a x}{\sqrt {b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^2}-\frac {5 \left (b c+a^2 d\right ) \arctan \left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \left (b c+a^2 d\right ) \text {arctanh}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 (-1)^{3/4} \left (b c+a^2 d\right ) \text {arctanh}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \sqrt [4]{-1} \left (b c+a^2 d\right ) \text {arctanh}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(869\) vs. \(2(384)=768\).
Time = 1.09 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.26, number of steps used = 35, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6874, 2147, 294, 335, 307, 217, 1179, 642, 1176, 631, 210, 218, 212, 209, 2145, 474, 470} \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {4 c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {5 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a^3}+\frac {5 d \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}-\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}+\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}} \]
[In]
[Out]
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 294
Rule 307
Rule 335
Rule 470
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2145
Rule 2147
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}}+\frac {c x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}}\right ) \, dx \\ & = c \int \frac {x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx \\ & = \frac {c \text {Subst}\left (\int \frac {\sqrt [4]{x} \left (b+x^2\right )^2}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^3}+\frac {(4 d) \text {Subst}\left (\int \frac {x^{9/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a} \\ & = \frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {c \text {Subst}\left (\int \frac {\sqrt [4]{x} \left (3 b^2+2 b x^2\right )}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^3 b}+\frac {(5 d) \text {Subst}\left (\int \frac {\sqrt [4]{x}}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(5 b c) \text {Subst}\left (\int \frac {\sqrt [4]{x}}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^3}+\frac {(10 d) \text {Subst}\left (\int \frac {x^4}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(10 b c) \text {Subst}\left (\int \frac {x^4}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}-\frac {(5 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a}+\frac {(5 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(5 b c) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}+\frac {(5 b c) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}-\frac {(5 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 )}{2 a \sqrt [4]{b}}-\frac {(5 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 )}{2 a \sqrt [4]{b}}+\frac {(5 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 )}{2 a \sqrt [4]{b}}+\frac {(5 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 )}{2 a \sqrt [4]{b}} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {\left (5 b^{3/4} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}-\frac {\left (5 b^{3/4} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}+\frac {\left (5 b^{3/4} c\right ) \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 )}{2 a^3}+\frac {\left (5 b^{3/4} c\right ) \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 )}{2 a^3}-\frac {(5 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 )}{4 \sqrt {2} a b^{3/8}}-\frac {(5 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 )}{4 \sqrt {2} a b^{3/8}}+\frac {(5 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 )}{4 a \sqrt [4]{b}}+\frac {(5 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 )}{4 a \sqrt [4]{b}} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 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 )}{4 \sqrt {2} a b^{3/8}}+\frac {5 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 )}{4 \sqrt {2} a b^{3/8}}-\frac {\left (5 b^{5/8} c\right ) \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 )}{4 \sqrt {2} a^3}-\frac {\left (5 b^{5/8} c\right ) \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 )}{4 \sqrt {2} a^3}+\frac {\left (5 b^{3/4} c\right ) \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 )}{4 a^3}+\frac {\left (5 b^{3/4} c\right ) \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 )}{4 a^3}+\frac {(5 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 )}{2 \sqrt {2} a b^{3/8}}-\frac {(5 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 )}{2 \sqrt {2} a b^{3/8}} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} 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 )}{4 \sqrt {2} a^3}-\frac {5 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 )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} 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 )}{4 \sqrt {2} a^3}+\frac {5 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 )}{4 \sqrt {2} a b^{3/8}}+\frac {\left (5 b^{5/8} c\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 )}{2 \sqrt {2} a^3}-\frac {\left (5 b^{5/8} c\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 )}{2 \sqrt {2} a^3} \\ & = \frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {5 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}+\frac {5 d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} 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 )}{4 \sqrt {2} a^3}-\frac {5 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 )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} 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 )}{4 \sqrt {2} a^3}+\frac {5 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 )}{4 \sqrt {2} a b^{3/8}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(13171\) vs. \(2(384)=768\).
Time = 40.29 (sec) , antiderivative size = 13171, normalized size of antiderivative = 34.30 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Result too large to show} \]
[In]
[Out]
\[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}{\left (a^{2} x^{2}-b \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 2872, normalized size of antiderivative = 7.48 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {5}{4}} \left (c x^{2} + d\right )}{\left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + d\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {5}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{5/4}\,\left (c\,x^2+d\right )}{{\left (a^2\,x^2-b\right )}^{3/2}} \,d x \]
[In]
[Out]