Integrand size = 18, antiderivative size = 609 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}} \]
-2/5*(a+b*arcsech(c*x))/e/(e*x+d)^(5/2)-16/15*b*c^3*EllipticE(1/2*(-c*x+1) ^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)* (e*x+d)^(1/2)/(c^2*d^2-e^2)^2/(c*(e*x+d)/(c*d+e))^(1/2)-4/5*b*c*EllipticE( 1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c *x+1)^(1/2)*(e*x+d)^(1/2)/d^2/(c^2*d^2-e^2)/(c*(e*x+d)/(c*d+e))^(1/2)+4/15 *b*c*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c *x+1))^(1/2)*(c*x+1)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)/d/(c^2*d^2-e^2)/(e*x+ d)^(1/2)+4/5*b*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e)) ^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)/d^2/e/(e *x+d)^(1/2)+4/15*b*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/ (c^2*d^2-e^2)/(e*x+d)^(3/2)+16/15*b*c^2*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)* (-c^2*x^2+1)^(1/2)/(c^2*d^2-e^2)^2/(e*x+d)^(1/2)+4/5*b*e*(1/(c*x+1))^(1/2) *(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/(c^2*d^2-e^2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 26.78 (sec) , antiderivative size = 8675, normalized size of antiderivative = 14.24 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Result too large to show} \]
Time = 0.88 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.71, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6842, 635, 632, 186, 413, 412, 688, 27, 688, 27, 600, 508, 327, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6842 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x (d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d^2}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d^2}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx-\frac {2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d^2}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}}dx-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 \int -\frac {e \left (3 d \left (2 c^2-\frac {e^2}{d^2}\right )-c^2 e x\right )}{2 d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \int \frac {3 \left (2 c^2 d-\frac {e^2}{d}\right )-c^2 e x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {2 \int \frac {c^2 \left (2 d \left (3 c^2 d^2-e^2\right )+e \left (7 c^2 d^2-3 e^2\right ) x\right )}{2 d \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \int \frac {2 d \left (3 c^2 d^2-e^2\right )+e \left (7 c^2 d^2-3 e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \left (\left (7 c^2 d^2-3 e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \left (-\frac {2 \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \left (-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \left (\frac {2 d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {c^2 \left (\frac {2 d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d \left (c^2 d^2-e^2\right )}+\frac {2 e \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (c^2 d^2-e^2\right )}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}\right )}{5 e}\) |
(-2*(a + b*ArcSech[c*x]))/(5*e*(d + e*x)^(5/2)) - (2*b*Sqrt[(1 + c*x)^(-1) ]*Sqrt[1 + c*x]*((-2*e^2*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d^2 - e^2)*(d + e*x) ^(3/2)) - (e*((2*e*(7*c^2*d^2 - 3*e^2)*Sqrt[1 - c^2*x^2])/(d*(c^2*d^2 - e^ 2)*Sqrt[d + e*x]) + (c^2*((-2*(7*c^2*d^2 - 3*e^2)*Sqrt[d + e*x]*EllipticE[ ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/(c* d + e)]) + (2*d*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Elliptic F[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x])))/(d* (c^2*d^2 - e^2))))/(3*d*(c^2*d^2 - e^2)) - (2*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(d^2* Sqrt[d + e/c - (e*(1 - c*x))/c])))/(5*e)
3.1.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] + Int[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x)^(m + 1)/(x* Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1611\) vs. \(2(539)=1078\).
Time = 14.25 (sec) , antiderivative size = 1612, normalized size of antiderivative = 2.65
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1612\) |
| default | \(\text {Expression too large to display}\) | \(1612\) |
| parts | \(\text {Expression too large to display}\) | \(1634\) |
2/e*(-1/5*a/(e*x+d)^(5/2)+b*(-1/5/(e*x+d)^(5/2)*arcsech(c*x)-2/15*c*e^2*(( -c*(e*x+d)+c*d+e)/c/e/x)^(1/2)*x*(-(-c*(e*x+d)+c*d-e)/c/e/x)^(1/2)*(6*((-c *(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*Elliptic F((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*c^4*d^4*(e*x+d) ^(3/2)-7*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^( 1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*c^ 4*d^4*(e*x+d)^(3/2)+3*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d- e)/(c*d-e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),1/c*(c*d+e)/d ,(c/(c*d-e))^(1/2)/(c/(c*d+e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)-7*(c/(c*d+e))^ (1/2)*c^4*d^3*(e*x+d)^3-7*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+ c*d-e)/(c*d-e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/( c*d-e))^(1/2))*c^3*d^3*e*(e*x+d)^(3/2)+7*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2 )*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d+e))^( 1/2),((c*d+e)/(c*d-e))^(1/2))*c^3*d^3*e*(e*x+d)^(3/2)+13*(c/(c*d+e))^(1/2) *c^4*d^4*(e*x+d)^2-2*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e )/(c*d-e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e ))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)+3*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*( (-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d+e))^(1/2 ),((c*d+e)/(c*d-e))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)-6*((-c*(e*x+d)+c*d+e) /(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticPi((e*x+d)^(...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
integral(sqrt(e*x + d)*(b*arcsech(c*x) + a)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2 *e^2*x^2 + 4*d^3*e*x + d^4), x)
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]