\(\int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {a+b x}} \, dx\) [445]

Optimal result

Integrand size = 26, antiderivative size = 373 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x}}{15 a^2 c^2 e^3 \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{15 a^2 c e^2 (e x)^{3/2}}-\frac {2 \sqrt {b} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {b} d (4 b c-a d) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{15 a^{3/2} c^2 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

-2/15*(-2*a^2*d^2-3*a*b*c*d+8*b^2*c^2)*(d*x+c)^(1/2)/a^2/c^2/e^3/(e*x)^(1/ 
2)/(b*x+a)^(1/2)-2/5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/e/(e*x)^(5/2)+2/15*(-a* 
d+4*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c/e^2/(e*x)^(3/2)-2/15*b^(1/2)*(- 
2*a^2*d^2-3*a*b*c*d+8*b^2*c^2)*(d*x+c)^(1/2)*EllipticE(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^(5/2)/c^2/e^(7/2)/(b 
*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+2/15*b^(1/2)*d*(-a*d+4*b*c)*(d*x+c 
)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a*d 
/b/c)^(1/2))/a^(3/2)/c^2/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 = 11.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {a+b x}} \, dx=\frac {x \left (-2 a c (a+b x) (c+d x) (3 a c-4 b c x+a d x)-2 i \sqrt {\frac {a}{b}} b d \left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+4 i \sqrt {\frac {a}{b}} b d \left (-2 b^2 c^2+a b c d+a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{15 a^3 c^2 (e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

(x*(-2*a*c*(a + b*x)*(c + d*x)*(3*a*c - 4*b*c*x + a*d*x) - (2*I)*Sqrt[a/b] 
*b*d*(-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*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)] + (4*I)*S 
qrt[a/b]*b*d*(-2*b^2*c^2 + a*b*c*d + a^2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c 
/(d*x)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)]))/(15 
*a^3*c^2*(e*x)^(7/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {110, 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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(5*a*e*(e*x)^(5/2)) - ((-2*(4*b*c - a*d)* 
Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)) - ((-2*(8*b^2*c^2 - 3*a 
*b*c*d - 2*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]) + (b*d* 
((2*Sqrt[-a]*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*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]) - (2*Sqrt[-a]*c*( 
b*c - a*d)*(8*b*c + a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[Arc 
Sin[(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*c*e))/(3*a*c*e))/(5*a*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 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

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. \(639\) vs. \(2(318)=636\).

Time = 3.46 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.72

method	result	size
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{5 e^{4} a \,x^{3}}-\frac {2 \left (a d -4 b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{15 e^{4} a^{2} c \,x^{2}}+\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right )}{15 e^{4} a^{3} c^{2} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (-\frac {b d \left (a d -4 b c \right )}{15 e^{3} a^{2} c}+\frac {\left (a d +b c \right ) \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right )}{15 a^{3} c^{2} e^{3}}-\frac {\left (a d e +b c e \right ) \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right )}{15 e^{4} a^{3} c^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}-\frac {2 b \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right ) \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{15 a^{3} c \,e^{3} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(640\)
default	\(\frac {2 \sqrt {x d +c}\, \sqrt {b x +a}\, \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3} x^{2}+2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2} x^{2}-4 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d \,x^{2}-2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3} x^{2}-\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2} x^{2}+11 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d \,x^{2}-8 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{3} c^{4} x^{2}+2 a^{2} b \,d^{4} x^{4}+3 a \,b^{2} c \,d^{3} x^{4}-8 b^{3} c^{2} d^{2} x^{4}+2 a^{3} d^{4} x^{3}+4 a^{2} b c \,d^{3} x^{3}-a \,b^{2} c^{2} d^{2} x^{3}-8 b^{3} c^{3} d \,x^{3}+a^{3} c \,d^{3} x^{2}+3 a^{2} b \,c^{2} d^{2} x^{2}-4 a \,b^{2} c^{3} d \,x^{2}-4 a^{3} c^{2} d^{2} x +a^{2} b \,c^{3} d x -3 c^{3} d \,a^{3}\right )}{15 x^{2} a^{3} d \,e^{3} \sqrt {e x}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) c^{2}}\)	\(778\)

Input:

int((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(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*(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^2/c*(a* 
d-4*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^3/c^2*(2*a^2*d^2+3*a*b*c*d-8*b^2*c^2)/(x*(b* 
d*e*x^2+a*d*e*x+b*c*e*x+a*c*e))^(1/2)+2*(-1/15*b*d/e^3*(a*d-4*b*c)/a^2/c+1 
/15*(a*d+b*c)*(2*a^2*d^2+3*a*b*c*d-8*b^2*c^2)/a^3/c^2/e^3-1/15*(a*d*e+b*c* 
e)/e^4/a^3/c^2*(2*a^2*d^2+3*a*b*c*d-8*b^2*c^2))*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)*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))-2/15* 
b*(2*a^2*d^2+3*a*b*c*d-8*b^2*c^2)/a^3/c/e^3*((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 [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} \sqrt {b d e} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (8 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} \sqrt {b d e} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (3 \, a^{2} b c^{2} d + {\left (8 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} - {\left (4 \, a b^{2} c^{2} d - a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{45 \, a^{3} b c^{2} d e^{4} x^{3}} \] Input:

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

Output:

-2/45*((8*b^3*c^3 - 7*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*a^3*d^3)*sqrt(b*d*e) 
*x^3*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/2 
7*(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*(8*b^3*c^2*d - 3*a*b^2*c*d^2 - 2*a^2*b*d^3 
)*sqrt(b*d*e)*x^3*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), 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*(3*a^2*b*c^2*d + (8*b^3*c^2*d - 3*a*b^2*c* 
d^2 - 2*a^2*b*d^3)*x^2 - (4*a*b^2*c^2*d - a^2*b*c*d^2)*x)*sqrt(b*x + a)*sq 
rt(d*x + c)*sqrt(e*x))/(a^3*b*c^2*d*e^4*x^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {a+b x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a \,x^{3}+\sqrt {x}\, b \,x^{4}}d x \right )}{e^{4}} \] Input:

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

Output:

(sqrt(e)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a*x**3 + sqrt(x)*b*x** 
4),x))/e**4