3.1.63 \(\int (d+e x)^{3/2} (a+b \sec ^{-1}(c x)) \, dx\) [63]

3.1.63.1 Optimal result

Integrand size = 18, antiderivative size = 372 \[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {28 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

output

2/5*(e*x+d)^(5/2)*(a+b*arcsec(c*x))/e+4/15*b*e*(-c^2*x^2+1)*(e*x+d)^(1/2)/ 
c^3/x/(1-1/c^2/x^2)^(1/2)+28/15*b*d*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2 
^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/x/(1-1/c^2/ 
x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+4/15*b*(2*c^2*d^2+e^2)*EllipticF(1/2* 
(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2 
)*(-c^2*x^2+1)^(1/2)/c^4/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/5*b*d^3*Ell 
ipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d) 
/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/c/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
 

3.1.63.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.35 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.90 \[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{15} \left (-\frac {4 b e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+\frac {6 a (d+e x)^{5/2}}{e}+\frac {6 b (d+e x)^{5/2} \sec ^{-1}(c x)}{e}+\frac {4 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (-7 c d (c d-e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+\left (9 c^2 d^2-7 c d e+e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-3 c^2 d^2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c^3 e \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right ) \]

input

Integrate[(d + e*x)^(3/2)*(a + b*ArcSec[c*x]),x]
 

output

((-4*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (6*a*(d + e*x)^(5/2))/ 
e + (6*b*(d + e*x)^(5/2)*ArcSec[c*x])/e + ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c 
*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-7*c*d*(c*d - e)*EllipticE[I*ArcSin 
h[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (9*c^2*d^2 - 
 7*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], ( 
c*d + e)/(c*d - e)] - 3*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c 
/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e*Sqrt[-(c/(c*d + 
 e))]*Sqrt[1 - 1/(c^2*x^2)]*x))/15
 

3.1.63.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {5749, 1898, 634, 633, 632, 186, 413, 412, 2185, 27, 600, 509, 508, 327, 512, 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.

\(\displaystyle \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 5749

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \int \frac {(d+e x)^{5/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{5 c e}\)

\(\Big \downarrow \) 1898

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {(d+e x)^{5/2}}{x \sqrt {x^2-\frac {1}{c^2}}}dx}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 634

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^3 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 633

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {d^3 \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 632

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {d^3 \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^3 \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{\sqrt {x^2-\frac {1}{c^2}}}-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 \int -\frac {e^3 \left (9 d^2 c^2+7 d e x c^2+e^2\right )}{2 c^2 \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 e^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \int \frac {9 d^2 c^2+7 d e x c^2+e^2}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+7 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {x^2-\frac {1}{c^2}}}dx\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {7 c^2 d \sqrt {1-c^2 x^2} \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {14 c d \sqrt {1-c^2 x^2} \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {14 c d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (\frac {\sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {14 c d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {14 c d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^3 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {14 c d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}+\frac {2}{3} e^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\)

input

Int[(d + e*x)^(3/2)*(a + b*ArcSec[c*x]),x]
 

output

(2*(d + e*x)^(5/2)*(a + b*ArcSec[c*x]))/(5*e) - (2*b*Sqrt[-c^(-2) + x^2]*( 
(2*e^2*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2])/3 + (e*((-14*c*d*Sqrt[d + e*x]*S 
qrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] 
)/(Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[-c^(-2) + x^2]) - (2*(2*c^2*d^2 + e^ 
2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 
 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2])) 
)/(3*c^2) - (2*d^3*Sqrt[1 - c^2*x^2]*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*Ell 
ipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[-c^(-2) 
+ x^2]*Sqrt[d + e/c - (e*(1 - c*x))/c])))/(5*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)
 

3.1.63.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 186

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]
 

rule 321

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])
 

rule 327

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]
 

rule 412

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])
 

rule 413

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]
 

rule 508

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]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

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]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

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]
 

rule 632

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]
 

rule 633

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 
+ b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 634

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[( 
1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n 
+ 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 1/2, 0]
 

rule 1898

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ 
(q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( 
c + a*x^(2*n))^FracPart[p])   Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n 
))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !I 
ntegerQ[p] &&  !IntegerQ[q] && PosQ[n]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

rule 5749

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSec[c*x])/(e*(m + 1))), x] - Simp[b/ 
(c*e*(m + 1))   Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / 
; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

3.1.63.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(797\) vs. \(2(335)=670\).

Time = 9.08 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.15

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsec}\left (c x \right )}{5}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+9 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-3 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}+7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(798\)
default	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsec}\left (c x \right )}{5}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+9 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-3 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}+7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(798\)
parts	\(\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5 e}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsec}\left (c x \right )}{5}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}+9 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}-7 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-3 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+7 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -7 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(812\)

input

int((e*x+d)^(3/2)*(a+b*arcsec(c*x)),x,method=_RETURNVERBOSE)
 

output

2/e*(1/5*a*(e*x+d)^(5/2)+b*(1/5*(e*x+d)^(5/2)*arcsec(c*x)-2/15/c^3*((c/(c* 
d-e))^(1/2)*c^2*(e*x+d)^(5/2)+9*d^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^2-7*((-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^2*d^2-3*d^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)*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^2-2*(c/(c*d-e))^(1/2)* 
c^2*d*(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)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+ 
e))^(1/2))*c*d*e-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*d*e+(c/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)^(1/2)+((-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))*e^2-(c/(c*d-e))^(1/2)*e^2*(e*x+ 
d)^(1/2))/(c/(c*d-e))^(1/2)/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2) 
/c^2/e^2/x^2)^(1/2)))
 

3.1.63.5 Fricas [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Timed out} \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsec(c*x)),x, algorithm="fricas")
 

output

Timed out
 

3.1.63.6 Sympy [F]

\[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]

input

integrate((e*x+d)**(3/2)*(a+b*asec(c*x)),x)
 

output

Integral((a + b*asec(c*x))*(d + e*x)**(3/2), x)
 

3.1.63.7 Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsec(c*x)),x, algorithm="maxima")
 

output

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)
 

3.1.63.8 Giac [F]

\[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \,d x } \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsec(c*x)),x, algorithm="giac")
 

output

integrate((e*x + d)^(3/2)*(b*arcsec(c*x) + a), x)
 

3.1.63.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]

input

int((a + b*acos(1/(c*x)))*(d + e*x)^(3/2),x)
 

output

int((a + b*acos(1/(c*x)))*(d + e*x)^(3/2), x)