\(\int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx\) [439]

Optimal result

Integrand size = 26, antiderivative size = 469 \[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{105 b^3 d^3 \sqrt {a+b x}}-\frac {2 \left (4 b^2 c^2+5 a b c d-24 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{105 b^3 d^2}+\frac {2 (b c-6 a d) e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{35 b^2 d}+\frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {2 \sqrt {a} \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) e^{5/2} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{105 b^{7/2} d^3 \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 a^{3/2} \left (4 b^2 c^2+5 a b c d-24 a^2 d^2\right ) e^{5/2} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{105 b^{7/2} d^2 \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2/105*(-48*a^3*d^3+16*a^2*b*c*d^2+9*a*b^2*c^2*d+8*b^3*c^3)*e^2*(e*x)^(1/2) 
*(d*x+c)^(1/2)/b^3/d^3/(b*x+a)^(1/2)-2/105*(-24*a^2*d^2+5*a*b*c*d+4*b^2*c^ 
2)*e^2*(e*x)^(1/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^2+2/35*(-6*a*d+b*c)*e 
*(e*x)^(3/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d+2/7*(e*x)^(5/2)*(b*x+a)^(1/ 
2)*(d*x+c)^(1/2)/b-2/105*a^(1/2)*(-48*a^3*d^3+16*a^2*b*c*d^2+9*a*b^2*c^2*d 
+8*b^3*c^3)*e^(5/2)*(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))/b^(7/2)/d^3/(b*x+a)^(1/2)/(a*(d*x 
+c)/c/(b*x+a))^(1/2)+2/105*a^(3/2)*(-24*a^2*d^2+5*a*b*c*d+4*b^2*c^2)*e^(5/ 
2)*(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))/b^(7/2)/d^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 = 16.12 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.80 \[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 (e x)^{5/2} \left (\frac {\left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) (a+b x) (c+d x)}{b \sqrt {x}}-d \sqrt {x} (a+b x) (c+d x) \left (-24 a^2 d^2+a b d (5 c+18 d x)+b^2 \left (4 c^2-3 c d x-15 d^2 x^2\right )\right )+i \sqrt {\frac {a}{b}} d \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-4 i \sqrt {\frac {a}{b}} d \left (b^3 c^3+a b^2 c^2 d+10 a^2 b c d^2-12 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{105 b^3 d^3 x^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

Integrate[((e*x)^(5/2)*Sqrt[c + d*x])/Sqrt[a + b*x],x]
 

Output:

(2*(e*x)^(5/2)*(((8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3) 
*(a + b*x)*(c + d*x))/(b*Sqrt[x]) - d*Sqrt[x]*(a + b*x)*(c + d*x)*(-24*a^2 
*d^2 + a*b*d*(5*c + 18*d*x) + b^2*(4*c^2 - 3*c*d*x - 15*d^2*x^2)) + I*Sqrt 
[a/b]*d*(8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3)*Sqrt[1 + 
 a/(b*x)]*Sqrt[1 + c/(d*x)]*x*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c 
)/(a*d)] - (4*I)*Sqrt[a/b]*d*(b^3*c^3 + a*b^2*c^2*d + 10*a^2*b*c*d^2 - 12* 
a^3*d^3)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x*EllipticF[I*ArcSinh[Sqrt[a/ 
b]/Sqrt[x]], (b*c)/(a*d)]))/(105*b^3*d^3*x^(5/2)*Sqrt[a + b*x]*Sqrt[c + d* 
x])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {112, 27, 171, 27, 171, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {2 \int \frac {e (e x)^{3/2} (5 a c-(b c-6 a d) x)}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \int \frac {(e x)^{3/2} (5 a c-(b c-6 a d) x)}{\sqrt {a+b x} \sqrt {c+d x}}dx}{7 b}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {2 \int \frac {e \sqrt {e x} \left (3 a c (b c-6 a d)+\left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \int \frac {\sqrt {e x} \left (3 a c (b c-6 a d)+\left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2 \int -\frac {e \left (a c \left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right )+\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}+\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \int \frac {a c \left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right )+\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \left (\frac {\left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}{7 b}-\frac {e \left (\frac {e \left (\frac {2}{3} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {e \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{3 b d}\right )}{5 b d}-\frac {2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}\right )}{7 b}\)

Input:

Int[((e*x)^(5/2)*Sqrt[c + d*x])/Sqrt[a + b*x],x]
 

Output:

(2*(e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(7*b) - (e*((-2*(b*c - 6*a*d)* 
(e*x)^(3/2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(5*b*d) + (e*((2*(5*a*c + (4*b*c^ 
2)/d - (24*a^2*d)/b)*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])/3 - (e*((2*Sqr 
t[-a]*(8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*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]) - 
(2*Sqrt[-a]*c*(b*c - a*d)*(8*b^2*c^2 + 13*a*b*c*d + 24*a^2*d^2)*Sqrt[1 + ( 
b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*S 
qrt[e])], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[c + d*x])))/ 
(3*b*d)))/(5*b*d)))/(7*b)
 

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 112

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + 
p + 1))), x] - Simp[1/(f*(m + n + p + 1))   Int[(a + b*x)^(m - 1)*(c + d*x) 
^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a 
*f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && 
GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && 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 [A] (verified)

Time = 1.22 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 e^{2} x^{2} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{7 b}+\frac {2 \left (c \,e^{3}-\frac {2 e^{2} \left (3 a d e +3 b c e \right )}{7 b}\right ) x \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{5 b d e}+\frac {2 \left (-\frac {5 e^{3} a c}{7 b}-\frac {2 \left (c \,e^{3}-\frac {2 e^{2} \left (3 a d e +3 b c e \right )}{7 b}\right ) \left (2 a d e +2 b c e \right )}{5 b d e}\right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 b d e}-\frac {2 \left (-\frac {5 e^{3} a c}{7 b}-\frac {2 \left (c \,e^{3}-\frac {2 e^{2} \left (3 a d e +3 b c e \right )}{7 b}\right ) \left (2 a d e +2 b c e \right )}{5 b d e}\right ) a \,c^{2} \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 )}{3 b \,d^{2} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (-\frac {3 \left (c \,e^{3}-\frac {2 e^{2} \left (3 a d e +3 b c e \right )}{7 b}\right ) a c}{5 b d}-\frac {2 \left (-\frac {5 e^{3} a c}{7 b}-\frac {2 \left (c \,e^{3}-\frac {2 e^{2} \left (3 a d e +3 b c e \right )}{7 b}\right ) \left (2 a d e +2 b c e \right )}{5 b d e}\right ) \left (a d e +b c e \right )}{3 b d e}\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 )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(720\)
default	\(\frac {2 e^{2} \sqrt {e x}\, \sqrt {x d +c}\, \sqrt {b x +a}\, \left (15 b^{4} d^{5} x^{5}+48 \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^{4} c \,d^{4}-40 \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} b \,c^{2} d^{3}-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^{2} b^{2} c^{3} d^{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^{3} c^{4} d -48 \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^{4} c \,d^{4}+64 \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} b \,c^{2} d^{3}-7 \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^{2} c^{3} d^{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 \,b^{3} c^{4} d -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^{4} c^{5}-3 a \,b^{3} d^{5} x^{4}+18 b^{4} c \,d^{4} x^{4}+6 a^{2} b^{2} d^{5} x^{3}-5 a \,b^{3} c \,d^{4} x^{3}-b^{4} c^{2} d^{3} x^{3}+24 a^{3} b \,d^{5} x^{2}+a^{2} b^{2} c \,d^{4} x^{2}-6 a \,b^{3} c^{2} d^{3} x^{2}-4 b^{4} c^{3} d^{2} x^{2}+24 a^{3} b c \,d^{4} x -5 a^{2} b^{2} c^{2} d^{3} x -4 a \,b^{3} c^{3} d^{2} x \right )}{105 x \left (b d \,x^{2}+a d x +b c x +a c \right ) b^{4} d^{4}}\)	\(926\)

Input:

int((e*x)^(5/2)*(d*x+c)^(1/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/e/x*(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*(e*x*(b*x+a)*(d*x+c))^(1/2)* 
(2/7*e^2/b*x^2*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)+2/5*(c*e^3-2/ 
7*e^2/b*(3*a*d*e+3*b*c*e))/b/d/e*x*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x) 
^(1/2)+2/3*(-5/7*e^3/b*a*c-2/5*(c*e^3-2/7*e^2/b*(3*a*d*e+3*b*c*e))/b/d/e*( 
2*a*d*e+2*b*c*e))/b/d/e*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)-2/3* 
(-5/7*e^3/b*a*c-2/5*(c*e^3-2/7*e^2/b*(3*a*d*e+3*b*c*e))/b/d/e*(2*a*d*e+2*b 
*c*e))/b/d^2*a*c^2*((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*(-3/5*(c*e^3-2/7*e^2/b*(3*a*d*e+3*b* 
c*e))/b/d*a*c-2/3*(-5/7*e^3/b*a*c-2/5*(c*e^3-2/7*e^2/b*(3*a*d*e+3*b*c*e))/ 
b/d/e*(2*a*d*e+2*b*c*e))/b/d/e*(a*d*e+b*c*e))*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 [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.04 \[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{4} c^{4} + 5 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 48 \, a^{4} d^{4}\right )} \sqrt {b d e} e^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (8 \, b^{4} c^{3} d + 9 \, a b^{3} c^{2} d^{2} + 16 \, a^{2} b^{2} c d^{3} - 48 \, a^{3} b d^{4}\right )} \sqrt {b d e} e^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (15 \, b^{4} d^{4} e^{2} x^{2} + 3 \, {\left (b^{4} c d^{3} - 6 \, a b^{3} d^{4}\right )} e^{2} x - {\left (4 \, b^{4} c^{2} d^{2} + 5 \, a b^{3} c d^{3} - 24 \, a^{2} b^{2} d^{4}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{315 \, b^{5} d^{4}} \] Input:

integrate((e*x)^(5/2)*(d*x+c)^(1/2)/(b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

-2/315*((8*b^4*c^4 + 5*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 - 
 48*a^4*d^4)*sqrt(b*d*e)*e^2*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*(8*b^4*c^3*d + 9*a 
*b^3*c^2*d^2 + 16*a^2*b^2*c*d^3 - 48*a^3*b*d^4)*sqrt(b*d*e)*e^2*weierstras 
sZeta(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*(15*b^4*d^4*e^2*x^2 + 3*(b^4*c*d^3 - 6*a*b^3*d^4)*e^2*x - (4*b^4*c^2*d 
^2 + 5*a*b^3*c*d^3 - 24*a^2*b^2*d^4)*e^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt 
(e*x))/(b^5*d^4)
 

Sympy [F]

\[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}} \sqrt {c + d x}}{\sqrt {a + b x}}\, dx \] Input:

integrate((e*x)**(5/2)*(d*x+c)**(1/2)/(b*x+a)**(1/2),x)
 

Output:

Integral((e*x)**(5/2)*sqrt(c + d*x)/sqrt(a + b*x), x)
 

Maxima [F]

\[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}}{\sqrt {b x + a}} \,d x } \] Input:

integrate((e*x)^(5/2)*(d*x+c)^(1/2)/(b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x + c)*(e*x)^(5/2)/sqrt(b*x + a), x)
 

Giac [F]

\[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}}{\sqrt {b x + a}} \,d x } \] Input:

integrate((e*x)^(5/2)*(d*x+c)^(1/2)/(b*x+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x + c)*(e*x)^(5/2)/sqrt(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{\sqrt {a+b\,x}} \,d x \] Input:

int(((e*x)^(5/2)*(c + d*x)^(1/2))/(a + b*x)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(((e*x)^(5/2)*(c + d*x)^(1/2))/(a + b*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^{5/2} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

int((e*x)^(5/2)*(d*x+c)^(1/2)/(b*x+a)^(1/2),x)
 

Output:

(sqrt(e)*e**2*(36*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*a**2*c*d - 24*sqrt(x 
)*sqrt(c + d*x)*sqrt(a + b*x)*a**2*d**2*x - 6*sqrt(x)*sqrt(c + d*x)*sqrt(a 
 + b*x)*a*b*c**2 - 20*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c*d*x + 20*s 
qrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**2*x**2 + 4*sqrt(x)*sqrt(c + d*x) 
*sqrt(a + b*x)*b**2*c**2*x + 20*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c 
*d*x**2 + 48*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2* 
d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x 
**2),x)*a**4*d**4 + 32*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c 
*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + 
b**2*c*d*x**2),x)*a**3*b*c*d**3 - 25*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b 
*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + 
b**2*c**2*x + b**2*c*d*x**2),x)*a**2*b**2*c**2*d**2 - 17*int((sqrt(x)*sqrt 
(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d* 
x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a*b**3*c**3*d - 8*int( 
(sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 
 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*b**4*c**4 
 - 18*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a**2*c*d*x + a**2*d**2*x* 
*2 + a*b*c**2*x + 2*a*b*c*d*x**2 + a*b*d**2*x**3 + b**2*c**2*x**2 + b**2*c 
*d*x**3),x)*a**4*c**2*d**2 - 15*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/ 
(a**2*c*d*x + a**2*d**2*x**2 + a*b*c**2*x + 2*a*b*c*d*x**2 + a*b*d**2*x...