\(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [195]

Optimal result

Integrand size = 23, antiderivative size = 301 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 (8 c d-5 b e) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}+\frac {4 c x \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {16 \sqrt {d} (2 c d-b e) \sqrt {b x+c x^2} E\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right )|1-\frac {c d}{b e}\right )}{3 e^{7/2} \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}}-\frac {2 \sqrt {d} (8 c d-3 b e) \sqrt {b x+c x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right ),1-\frac {c d}{b e}\right )}{3 e^{7/2} \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}} \] Output:

-2/3*(-5*b*e+8*c*d)*(c*x^2+b*x)^(1/2)/e^3/(e*x+d)^(1/2)+4/3*c*x*(c*x^2+b*x 
)^(1/2)/e^2/(e*x+d)^(1/2)-2/3*(c*x^2+b*x)^(3/2)/e/(e*x+d)^(3/2)+16/3*d^(1/ 
2)*(-b*e+2*c*d)*(c*x^2+b*x)^(1/2)*EllipticE(e^(1/2)*x^(1/2)/d^(1/2)/(1+e*x 
/d)^(1/2),(1-c*d/b/e)^(1/2))/e^(7/2)/x^(1/2)/(d*(c*x+b)/b/(e*x+d))^(1/2)/( 
e*x+d)^(1/2)-2/3*d^(1/2)*(-3*b*e+8*c*d)*(c*x^2+b*x)^(1/2)*InverseJacobiAM( 
arctan(e^(1/2)*x^(1/2)/d^(1/2)),(1-c*d/b/e)^(1/2))/e^(7/2)/x^(1/2)/(d*(c*x 
+b)/b/(e*x+d))^(1/2)/(e*x+d)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (x (b+c x))^{3/2} \left (8 (-2 c d+b e) (b+c x) (d+e x)+\frac {e x (b+c x) \left (-b e (3 d+4 e x)+c \left (8 d^2+10 d e x+e^2 x^2\right )\right )}{d+e x}+8 i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e (-8 c d+5 b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{3 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \] Input:

Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]
 

Output:

(2*(x*(b + c*x))^(3/2)*(8*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (e*x*(b + c 
*x)*(-(b*e*(3*d + 4*e*x)) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(d + e*x) + ( 
8*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3 
/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e 
*(-8*c*d + 5*b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I* 
ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*e^4*x^2*(b + c*x)^2*Sqrt[d + 
 e*x])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1161, 1230, 27, 1269, 1169, 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.

\(\displaystyle \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1169

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[(b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]
 

Output:

(-2*(b*x + c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + ((2*(8*c*d - 3*b*e + 2*c* 
e*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((16*Sqrt[-b]*Sqrt[c]*(2*c 
*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c 
]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2] 
) - (2*Sqrt[-b]*(4*c*d - 3*b*e)*(4*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sq 
rt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)] 
)/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*e^2))/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 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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[m^2, 1/4]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(580\) vs. \(2(256)=512\).

Time = 2.46 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.93

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{5} \left (x +\frac {d}{e}\right )^{2}}-\frac {8 \left (c e \,x^{2}+b e x \right ) \left (b e -2 c d \right )}{3 e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{3}}+\frac {2 \left (\frac {b^{2} e^{2}-4 b c d e +3 c^{2} d^{2}}{e^{4}}+\frac {d \left (b e -c d \right ) c}{3 e^{4}}-\frac {4 \left (b e -2 c d \right ) \left (b e -c d \right )}{3 e^{4}}+\frac {4 b \left (b e -2 c d \right )}{3 e^{3}}-\frac {c b d}{3 e^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {2 c \left (b e -c d \right )}{e^{3}}+\frac {4 \left (b e -2 c d \right ) c}{3 e^{3}}-\frac {2 c \left (b e +c d \right )}{3 e^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(581\)
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (5 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{3} x -8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{2} e^{2} x -8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{3} x +24 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{2} e^{2} x -16 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{3} e x +5 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d^{2} e^{2}-8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{3} e -8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d^{2} e^{2}+24 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{3} e -16 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{4}-c^{2} x^{4} e^{4}+3 x^{3} b c \,e^{4}-10 d \,c^{2} x^{3} e^{3}+4 x^{2} b^{2} e^{4}-7 x^{2} b c d \,e^{3}-8 x^{2} c^{2} d^{2} e^{2}+3 b^{2} d \,e^{3} x -8 b c \,d^{2} e^{2} x \right )}{3 \left (c x +b \right ) x \,e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(891\)

Input:

int((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)*x*(e*x+d))^(1/2)/x/(c*x+b)*(2/3 
*d*(b*e-c*d)/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2-8/3*(c*e* 
x^2+b*e*x)*(b*e-2*c*d)/e^4/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/3*c/e^3*(c*e* 
x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*((b^2*e^2-4*b*c*d*e+3*c^2*d^2)/e^4+1/3* 
d*(b*e-c*d)/e^4*c-4/3*(b*e-2*c*d)/e^4*(b*e-c*d)+4/3*b/e^3*(b*e-2*c*d)-1/3* 
c/e^3*b*d)*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/ 
2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d 
/e/(-d/e+b/c))^(1/2))+2*(2*c/e^3*(b*e-c*d)+4/3*(b*e-2*c*d)/e^3*c-2/3*c/e^3 
*(b*e+c*d))*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1 
/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e)/d 
*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e 
/(-d/e+b/c))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (256) = 512\).

Time = 0.17 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.76 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left ({\left (16 \, c^{2} d^{4} - 16 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (16 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 2 \, {\left (16 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (c^{2} e^{4} x^{2} + 8 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + 2 \, {\left (5 \, c^{2} d e^{3} - 2 \, b c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (c e^{7} x^{2} + 2 \, c d e^{6} x + c d^{2} e^{5}\right )}} \] Input:

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x, algorithm="fricas")
 

Output:

2/9*((16*c^2*d^4 - 16*b*c*d^3*e + b^2*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d 
*e^3 + b^2*e^4)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqr 
t(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4 
/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3 
*(3*c*e*x + c*d + b*e)/(c*e)) + 24*(2*c^2*d^3*e - b*c*d^2*e^2 + (2*c^2*d*e 
^3 - b*c*e^4)*x^2 + 2*(2*c^2*d^2*e^2 - b*c*d*e^3)*x)*sqrt(c*e)*weierstrass 
Zeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c 
^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*( 
c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 
 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 
 3*(c^2*e^4*x^2 + 8*c^2*d^2*e^2 - 3*b*c*d*e^3 + 2*(5*c^2*d*e^3 - 2*b*c*e^4 
)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c*e^7*x^2 + 2*c*d*e^6*x + c*d^2*e^5 
)
 

Sympy [F]

\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(5/2),x)
 

Output:

Integral((x*(b + c*x))**(3/2)/(d + e*x)**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int((b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x)
 

Output:

int((b*x + c*x^2)^(3/2)/(d + e*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\text {too large to display} \] Input:

int((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x)
 

Output:

(12*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*d*e + 8*sqrt(x)*sqrt(d + e*x) 
*sqrt(b + c*x)*b**2*e**2*x - 18*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c*d* 
*2 - 20*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c*d*e*x + 2*sqrt(x)*sqrt(d + 
 e*x)*sqrt(b + c*x)*b*c*e**2*x**2 + 12*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x) 
*c**2*d**2*x - 2*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**2*d*e*x**2 - 6*int 
((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**2*d**3*e + 3*sqrt(x)*b**2*d**2* 
e**2*x + 3*sqrt(x)*b**2*d*e**3*x**2 + sqrt(x)*b**2*e**4*x**3 - sqrt(x)*b*c 
*d**4 - 2*sqrt(x)*b*c*d**3*e*x + 2*sqrt(x)*b*c*d*e**3*x**3 + sqrt(x)*b*c*e 
**4*x**4 - sqrt(x)*c**2*d**4*x - 3*sqrt(x)*c**2*d**3*e*x**2 - 3*sqrt(x)*c* 
*2*d**2*e**2*x**3 - sqrt(x)*c**2*d*e**3*x**4),x)*b**4*d**4*e**2 - 12*int(( 
sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**2*d**3*e + 3*sqrt(x)*b**2*d**2*e* 
*2*x + 3*sqrt(x)*b**2*d*e**3*x**2 + sqrt(x)*b**2*e**4*x**3 - sqrt(x)*b*c*d 
**4 - 2*sqrt(x)*b*c*d**3*e*x + 2*sqrt(x)*b*c*d*e**3*x**3 + sqrt(x)*b*c*e** 
4*x**4 - sqrt(x)*c**2*d**4*x - 3*sqrt(x)*c**2*d**3*e*x**2 - 3*sqrt(x)*c**2 
*d**2*e**2*x**3 - sqrt(x)*c**2*d*e**3*x**4),x)*b**4*d**3*e**3*x - 6*int((s 
qrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**2*d**3*e + 3*sqrt(x)*b**2*d**2*e** 
2*x + 3*sqrt(x)*b**2*d*e**3*x**2 + sqrt(x)*b**2*e**4*x**3 - sqrt(x)*b*c*d* 
*4 - 2*sqrt(x)*b*c*d**3*e*x + 2*sqrt(x)*b*c*d*e**3*x**3 + sqrt(x)*b*c*e**4 
*x**4 - sqrt(x)*c**2*d**4*x - 3*sqrt(x)*c**2*d**3*e*x**2 - 3*sqrt(x)*c**2* 
d**2*e**2*x**3 - sqrt(x)*c**2*d*e**3*x**4),x)*b**4*d**2*e**4*x**2 + 15*...