\(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [145]

Optimal result

Integrand size = 21, antiderivative size = 219 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 (4 c d-b e) \sqrt {b x+c x^2}}{4 d e^3}-\frac {3 (2 c d-b e) x \sqrt {b x+c x^2}}{4 d e^2 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{4 \sqrt {d} e^4 \sqrt {c d-b e}} \] Output:

3/4*(-b*e+4*c*d)*(c*x^2+b*x)^(1/2)/d/e^3-3/4*(-b*e+2*c*d)*x*(c*x^2+b*x)^(1 
/2)/d/e^2/(e*x+d)-1/2*(c*x^2+b*x)^(3/2)/e/(e*x+d)^2-3*c^(1/2)*(-b*e+2*c*d) 
*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/e^4+3/4*(b^2*e^2-8*b*c*d*e+8*c^2*d^2 
)*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(1/2)/e^4/(-b*e+ 
c*d)^(1/2)
 

Mathematica [A] (verified)

Time = 11.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.08 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {e (-c d+b e) \sqrt {x} \left (-b e (3 d+5 e x)+2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )\right )}{(d+e x)^2}+\frac {12 \sqrt {c} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}-\frac {3 \sqrt {c d-b e} \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {d} \sqrt {b+c x}}\right )}{4 e^4 (-c d+b e) \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(b + c*x)]*((e*(-(c*d) + b*e)*Sqrt[x]*(-(b*e*(3*d + 5*e*x)) + 2*c* 
(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (12*Sqrt[c]*(2*c^2*d^2 - 3*b 
*c*d*e + b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c 
*x)/b]) - (3*Sqrt[c*d - b*e]*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sq 
rt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[b + c*x]))) 
/(4*e^4*(-(c*d) + b*e)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1161, 1230, 1269, 1091, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*(b*x + c*x^2)^(3/2)/(e*(d + e*x)^2) + (3*(((4*c*d - b*e + 2*c*e*x)*Sq 
rt[b*x + c*x^2])/(e^2*(d + e*x)) - ((8*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt 
[c]*x)/Sqrt[b*x + c*x^2]])/e - ((8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[ 
(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(S 
qrt[d]*e*Sqrt[c*d - b*e]))/(2*e^2)))/(4*e)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

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 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 [A] (verified)

Time = 0.89 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(-\frac {2 \left (3 \left (e x +d \right )^{2} x \sqrt {c}\, \left (c^{2} d^{2}-b c d e +\frac {1}{8} b^{2} e^{2}\right ) b \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (-\frac {3 b c x \left (e x +d \right )^{2} \left (b e -2 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{2}+e \sqrt {c}\, \left (\left (-c \,d^{2}+\frac {3}{8} b d e \right ) \left (x \left (c x +b \right )\right )^{\frac {3}{2}}+\frac {5 \left (-\frac {4 c \left (-2 c x +b \right ) d^{2}}{5}-\frac {21 b c d e x}{5}+b \,e^{2} x \left (-\frac {4 c x}{5}+b \right )\right ) x \sqrt {x \left (c x +b \right )}}{8}\right )\right )\right )}{\sqrt {c}\, \sqrt {d \left (b e -c d \right )}\, b \,e^{4} x \left (e x +d \right )^{2}}\)	\(221\)
risch	\(\frac {x \left (c x +b \right ) c}{e^{3} \sqrt {x \left (c x +b \right )}}+\frac {-\frac {2 \left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {3 \sqrt {c}\, \left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e}-\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}+\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 \left (b e -2 c d \right ) e \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{4}}}{2 e^{3}}\)	\(970\)
default	\(\text {Expression too large to display}\)	\(1628\)

Input:

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

Output:

-2/c^(1/2)*(3*(e*x+d)^2*x*c^(1/2)*(c^2*d^2-b*c*d*e+1/8*b^2*e^2)*b*arctan(( 
x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(-3/2*b*c*x* 
(e*x+d)^2*(b*e-2*c*d)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+e*c^(1/2)*((-c* 
d^2+3/8*b*d*e)*(x*(c*x+b))^(3/2)+5/8*(-4/5*c*(-2*c*x+b)*d^2-21/5*b*c*d*e*x 
+b*e^2*x*(-4/5*c*x+b))*x*(x*(c*x+b))^(1/2))))/(d*(b*e-c*d))^(1/2)/b/e^4/x/ 
(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (189) = 378\).

Time = 0.14 (sec) , antiderivative size = 1745, normalized size of antiderivative = 7.97 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(12*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c* 
d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x 
)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 3*(8*c^2*d^4 - 8* 
b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2* 
(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d 
+ (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 
2*(12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d* 
e^4)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + 
 b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^3*e^5 - 
 b*d^2*e^6)*x), -1/4*(3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^ 
2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2* 
d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b* 
x)/(c*d*x + b*d)) + 6*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3* 
e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^ 
2*d^2*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (12*c 
^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)*x^ 
2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/ 
(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^3*e^5 - b*d^2* 
e^6)*x), 1/8*(24*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 
 3*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (189) = 378\).

Time = 0.16 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.27 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {c x^{2} + b x} c}{e^{3}} + \frac {3 \, {\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, \sqrt {-c d^{2} + b d e} e^{4}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2 \, \sqrt {c} e^{4}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2} + 10 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, b^{3} \sqrt {c} d^{2} e}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2} e^{4}} \] Input:

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

Output:

sqrt(c*x^2 + b*x)*c/e^3 + 3/4*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*arctan(-(( 
sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/(sqrt( 
-c*d^2 + b*d*e)*e^4) + 3/2*(2*c^2*d - b*c*e)*log(abs(2*(sqrt(c)*x - sqrt(c 
*x^2 + b*x))*sqrt(c) + b))/(sqrt(c)*e^4) + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 
 + b*x))^3*c^2*d^2*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 + 5* 
(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b 
*x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e 
- (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)*d*e^2 + 40*(sqrt(c)*x - sq 
rt(c*x^2 + b*x))*b*c^2*d^3 - 28*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2* 
e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^2 + 10*b^2*c^(3/2)*d^3 - 5*b 
^3*sqrt(c)*d^2*e)/(((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - s 
qrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2*e^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 2312, normalized size of antiderivative = 10.56 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c* 
x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**3*d**2*e**3 - 24*sqrt( 
d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x) 
*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**3*d*e**4*x - 12*sqrt(d)*sqrt(b*e - 
 c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt 
(c))/(sqrt(d)*sqrt(c)))*b**3*e**5*x**2 + 120*sqrt(d)*sqrt(b*e - c*d)*atan( 
(sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt( 
d)*sqrt(c)))*b**2*c*d**3*e**2 + 240*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e 
 - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c 
)))*b**2*c*d**2*e**3*x + 120*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) 
 - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b** 
2*c*d*e**4*x**2 - 288*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt 
(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d** 
4*e - 576*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + 
 c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d**3*e**2*x - 2 
88*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - 
 sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e**3*x**2 + 192*s 
qrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqr 
t(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c**3*d**5 + 384*sqrt(d)*sqrt(b*e 
- c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*...