\(\int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [474]

Optimal result

Integrand size = 26, antiderivative size = 424 \[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}+\frac {4 (3 b c+2 a d) \sqrt {c+d x}}{15 a^2 c^2 e^2 (e x)^{3/2} \sqrt {a+b x}}-\frac {2 \left (24 b^2 c^2+13 a b c d+8 a^2 d^2\right ) \sqrt {c+d x}}{15 a^3 c^3 e^3 \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {b} \left (48 b^3 c^3-16 a b^2 c^2 d-9 a^2 b c d^2-8 a^3 d^3\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{15 a^{7/2} c^3 (b c-a d) e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {b} d \left (24 b^2 c^2-5 a b c d-4 a^2 d^2\right ) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 (b c-a d) e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

-2/5*(d*x+c)^(1/2)/a/c/e/(e*x)^(5/2)/(b*x+a)^(1/2)+4/15*(2*a*d+3*b*c)*(d*x 
+c)^(1/2)/a^2/c^2/e^2/(e*x)^(3/2)/(b*x+a)^(1/2)-2/15*(8*a^2*d^2+13*a*b*c*d 
+24*b^2*c^2)*(d*x+c)^(1/2)/a^3/c^3/e^3/(e*x)^(1/2)/(b*x+a)^(1/2)-2/15*b^(1 
/2)*(-8*a^3*d^3-9*a^2*b*c*d^2-16*a*b^2*c^2*d+48*b^3*c^3)*(d*x+c)^(1/2)*Ell 
ipticE(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/ 
2))/a^(7/2)/c^3/(-a*d+b*c)/e^(7/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/ 
2)+2/15*b^(1/2)*d*(-4*a^2*d^2-5*a*b*c*d+24*b^2*c^2)*(d*x+c)^(1/2)*InverseJ 
acobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a*d/b/c)^(1/2))/a^( 
5/2)/c^3/(-a*d+b*c)/e^(7/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.75 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {x \left (-2 a c (c+d x) \left (24 b^3 c^2 x^2+a b^2 c x (6 c-5 d x)+a^3 d (3 c-4 d x)-a^2 b \left (3 c^2+2 c d x+4 d^2 x^2\right )\right )+2 i \sqrt {\frac {a}{b}} b d \left (-48 b^3 c^3+16 a b^2 c^2 d+9 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b d \left (-24 b^3 c^3+11 a b^2 c^2 d+5 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{15 a^4 c^3 (-b c+a d) (e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(7/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

(x*(-2*a*c*(c + d*x)*(24*b^3*c^2*x^2 + a*b^2*c*x*(6*c - 5*d*x) + a^3*d*(3* 
c - 4*d*x) - a^2*b*(3*c^2 + 2*c*d*x + 4*d^2*x^2)) + (2*I)*Sqrt[a/b]*b*d*(- 
48*b^3*c^3 + 16*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 8*a^3*d^3)*Sqrt[1 + a/(b*x)] 
*Sqrt[1 + c/(d*x)]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/( 
a*d)] - (2*I)*Sqrt[a/b]*b*d*(-24*b^3*c^3 + 11*a*b^2*c^2*d + 5*a^2*b*c*d^2 
+ 8*a^3*d^3)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(7/2)*EllipticF[I*ArcSi 
nh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)]))/(15*a^4*c^3*(-(b*c) + a*d)*(e*x)^(7/ 
2)*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {115, 27, 169, 27, 169, 27, 169, 27, 176, 122, 120, 127, 126}

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.

\(\displaystyle \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {2 \int \frac {e \left (24 b^2 c^2+13 a b d c+8 a^2 d^2+6 b d (3 b c+2 a d) x\right )}{2 (e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e^2}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {24 b^2 c^2+13 a b d c+8 a^2 d^2+6 b d (3 b c+2 a d) x}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {b e \left (4 c \left (12 b^2 c^2+2 a b d c+a^2 d^2\right )+d \left (24 b^2 c^2+13 a b d c+8 a^2 d^2\right ) x\right )}{2 \sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e^2}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {b \int \frac {4 c \left (12 b^2 c^2+2 a b d c+a^2 d^2\right )+d \left (24 b^2 c^2+13 a b d c+8 a^2 d^2\right ) x}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \int -\frac {d e \left (a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )+\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a e (b c-a d)}+\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \int \frac {a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )+\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 176

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 122

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 120

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 127

\(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

\(\Big \downarrow \) 126

\(\displaystyle -\frac {-\frac {-\frac {2 \sqrt {c+d x} \left (8 a^2 d^2+13 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {8 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c e}}{3 a c e}-\frac {4 \sqrt {c+d x} (2 a d+3 b c)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a c e}-\frac {2 \sqrt {c+d x}}{5 a c e (e x)^{5/2} \sqrt {a+b x}}\)

Input:

Int[1/((e*x)^(7/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

(-2*Sqrt[c + d*x])/(5*a*c*e*(e*x)^(5/2)*Sqrt[a + b*x]) - ((-4*(3*b*c + 2*a 
*d)*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)*Sqrt[a + b*x]) - ((-2*(24*b^2*c^2 
+ 13*a*b*c*d + 8*a^2*d^2)*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]*Sqrt[a + b*x]) - 
 (b*((2*(48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*Sqrt[e*x 
]*Sqrt[c + d*x])/(a*(b*c - a*d)*e*Sqrt[a + b*x]) - (d*((2*Sqrt[-a]*(48*b^3 
*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*Sqrt[1 + (b*x)/a]*Sqrt[ 
c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/( 
b*c)])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (8*Sqrt[-a]*c 
*(b*c - a*d)*(12*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x)/a]*Sqrt[1 + 
 (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/ 
(b*c)])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[c + d*x])))/(a*(b*c - a*d))) 
)/(a*c*e))/(3*a*c*e))/(5*a*c*e)
 

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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(764\) vs. \(2(369)=738\).

Time = 5.53 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{5 e^{4} a^{2} c \,x^{3}}+\frac {2 \left (4 a d +9 b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{15 e^{4} a^{3} c^{2} x^{2}}-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right )}{15 e^{4} a^{4} c^{3} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (b d e \,x^{2}+b c e x \right ) b^{3}}{\left (a d -b c \right ) a^{4} e^{4} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (\frac {b d \left (4 a d +9 b c \right )}{15 e^{3} a^{3} c^{2}}-\frac {\left (a d +b c \right ) \left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right )}{15 a^{4} c^{3} e^{3}}+\frac {\left (a d e +b c e \right ) \left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right )}{15 e^{4} a^{4} c^{3}}-\frac {b^{3}}{a^{4} e^{3}}-\frac {b^{4} c}{e^{3} \left (a d -b c \right ) a^{4}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (\frac {d b \left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right )}{15 a^{4} c^{3} e^{3}}-\frac {b^{4} d}{\left (a d -b c \right ) a^{4} e^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(765\)
default	\(\text {Expression too large to display}\)	\(1016\)

Input:

int(1/(e*x)^(7/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(e*x*(b*x+a)*(d*x+c))^(1/2)/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*(-2/5/ 
e^4/a^2/c*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^3+2/15/e^4/a^3/c 
^2*(4*a*d+9*b*c)*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^2-2/15*(b 
*d*e*x^2+a*d*e*x+b*c*e*x+a*c*e)/e^4/a^4/c^3*(8*a^2*d^2+17*a*b*c*d+33*b^2*c 
^2)/(x*(b*d*e*x^2+a*d*e*x+b*c*e*x+a*c*e))^(1/2)+2*(b*d*e*x^2+b*c*e*x)/(a*d 
-b*c)*b^3/a^4/e^4/((x+a/b)*(b*d*e*x^2+b*c*e*x))^(1/2)+2*(1/15*b*d/e^3*(4*a 
*d+9*b*c)/a^3/c^2-1/15*(a*d+b*c)*(8*a^2*d^2+17*a*b*c*d+33*b^2*c^2)/a^4/c^3 
/e^3+1/15*(a*d*e+b*c*e)/e^4/a^4/c^3*(8*a^2*d^2+17*a*b*c*d+33*b^2*c^2)-b^3/ 
a^4/e^3-b^4*c/e^3/(a*d-b*c)/a^4)*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/ 
b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)*E 
llipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(1/15*d*b*(8*a^2*d 
^2+17*a*b*c*d+33*b^2*c^2)/a^4/c^3/e^3-b^4*d/(a*d-b*c)/a^4/e^3)*c/d*((x+c/d 
)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*e*x^3+a*d*e* 
x^2+b*c*e*x^2+a*c*e*x)^(1/2)*((-c/d+a/b)*EllipticE(((x+c/d)/c*d)^(1/2),(-c 
/d/(-c/d+a/b))^(1/2))-a/b*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^ 
(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (369) = 738\).

Time = 0.13 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x)^(7/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

-2/45*(3*(3*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + (48*b^5*c^3*d - 16*a*b^4*c^2 
*d^2 - 9*a^2*b^3*c*d^3 - 8*a^3*b^2*d^4)*x^3 + (24*a*b^4*c^3*d - 11*a^2*b^3 
*c^2*d^2 - 5*a^3*b^2*c*d^3 - 8*a^4*b*d^4)*x^2 - 2*(3*a^2*b^3*c^3*d - a^3*b 
^2*c^2*d^2 - 2*a^4*b*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x) + ((4 
8*b^5*c^4 - 40*a*b^4*c^3*d - 10*a^2*b^3*c^2*d^2 - 5*a^3*b^2*c*d^3 - 8*a^4* 
b*d^4)*x^4 + (48*a*b^4*c^4 - 40*a^2*b^3*c^3*d - 10*a^3*b^2*c^2*d^2 - 5*a^4 
*b*c*d^3 - 8*a^5*d^4)*x^3)*sqrt(b*d*e)*weierstrassPInverse(4/3*(b^2*c^2 - 
a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c 
*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d)) + 3*((48*b^5 
*c^3*d - 16*a*b^4*c^2*d^2 - 9*a^2*b^3*c*d^3 - 8*a^3*b^2*d^4)*x^4 + (48*a*b 
^4*c^3*d - 16*a^2*b^3*c^2*d^2 - 9*a^3*b^2*c*d^3 - 8*a^4*b*d^4)*x^3)*sqrt(b 
*d*e)*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*( 
2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weierstr 
assPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 
- 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c 
 + a*d)/(b*d))))/((a^4*b^3*c^4*d - a^5*b^2*c^3*d^2)*e^4*x^4 + (a^5*b^2*c^4 
*d - a^6*b*c^3*d^2)*e^4*x^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(7/2)/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(7/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*(e*x)^(7/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(7/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*(e*x)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{7/2}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/((e*x)^(7/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/((e*x)^(7/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{7/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(7/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x)
 

Output:

(sqrt(e)*( - 6*sqrt(c + d*x)*sqrt(a + b*x)*a*c + 8*sqrt(c + d*x)*sqrt(a + 
b*x)*a*d*x + 12*sqrt(c + d*x)*sqrt(a + b*x)*b*c*x + 30*sqrt(c + d*x)*sqrt( 
a + b*x)*b*d*x**2 + 15*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x 
)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + s 
qrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*d**2*x**2 + 15*sqrt(x) 
*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 
2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b 
**2*d*x**3),x)*b**3*d**2*x**3 + 8*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x) 
)/(sqrt(x)*a**2*c*x + sqrt(x)*a**2*d*x**2 + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt( 
x)*a*b*d*x**3 + sqrt(x)*b**2*c*x**3 + sqrt(x)*b**2*d*x**4),x)*a**3*d**2*x* 
*2 + 28*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c*x + sqrt 
(x)*a**2*d*x**2 + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**3 + sqrt(x)*b* 
*2*c*x**3 + sqrt(x)*b**2*d*x**4),x)*a**2*b*c*d*x**2 + 8*sqrt(x)*int((sqrt( 
c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c*x + sqrt(x)*a**2*d*x**2 + 2*sqrt(x 
)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**3 + sqrt(x)*b**2*c*x**3 + sqrt(x)*b**2*d 
*x**4),x)*a**2*b*d**2*x**3 + 24*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/ 
(sqrt(x)*a**2*c*x + sqrt(x)*a**2*d*x**2 + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x) 
*a*b*d*x**3 + sqrt(x)*b**2*c*x**3 + sqrt(x)*b**2*d*x**4),x)*a*b**2*c**2*x* 
*2 + 28*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c*x + sqrt 
(x)*a**2*d*x**2 + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**3 + sqrt(x)...