\(\int \frac {x (a+b x+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [84]

Optimal result

Integrand size = 23, antiderivative size = 414 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\left (b^2-4 a c\right ) \left (5 b^2 d e+28 a c d e-12 b \left (c d^2+a e^2\right )\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (5 b^2 d e+28 a c d e-12 b \left (c d^2+a e^2\right )\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}+\frac {d \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}+\frac {\left (2 c d^2+e (5 b d-12 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (5 b^2 d e+28 a c d e-12 b \left (c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^{9/2}} \] Output:

-1/512*(-4*a*c+b^2)*(5*b^2*d*e+28*a*c*d*e-12*b*(a*e^2+c*d^2))*(b*d-2*a*e+( 
-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)^2+1/192*( 
5*b^2*d*e+28*a*c*d*e-12*b*(a*e^2+c*d^2))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2 
+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^4+1/6*d*(c*x^2+b*x+a)^(5/2)/(a 
*e^2-b*d*e+c*d^2)/(e*x+d)^6+1/60*(2*c*d^2+e*(-12*a*e+5*b*d))*(c*x^2+b*x+a) 
^(5/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^5+1/1024*(-4*a*c+b^2)^2*(5*b^2*d*e+28 
*a*c*d*e-12*b*(a*e^2+c*d^2))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2 
-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)
 

Mathematica [A] (verified)

Time = 12.47 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.86 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {\frac {d (a+x (b+c x))^{5/2}}{(d+e x)^6}+\frac {\left (2 c d^2+e (5 b d-12 a e)\right ) (a+x (b+c x))^{5/2}}{10 \left (c d^2+e (-b d+a e)\right ) (d+e x)^5}-\frac {\left (\frac {5}{2} b^2 d e+14 a c d e-6 b \left (c d^2+a e^2\right )\right ) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{32 \left (c d^2+e (-b d+a e)\right )^2}}{6 \left (c d^2+e (-b d+a e)\right )} \] Input:

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

Output:

((d*(a + x*(b + c*x))^(5/2))/(d + e*x)^6 + ((2*c*d^2 + e*(5*b*d - 12*a*e)) 
*(a + x*(b + c*x))^(5/2))/(10*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - (( 
(5*b^2*d*e)/2 + 14*a*c*d*e - 6*b*(c*d^2 + a*e^2))*((2*(-(b*d) + 2*a*e - 2* 
c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sq 
rt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d 
) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x 
+ b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c* 
d^2 + e*(-(b*d) + a*e))^(3/2)))))/(32*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*(c 
*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1237, 27, 1228, 1152, 1152, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {(5 b d-2 c x d-12 a e) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)^6}dx}{6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {(5 b d-2 c x d-12 a e) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^6}dx}{12 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\left (-12 b \left (a e^2+c d^2\right )+28 a c d e+5 b^2 d e\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(d+e x)^5}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e (5 b d-12 a e)+2 c d^2\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\left (-12 b \left (a e^2+c d^2\right )+28 a c d e+5 b^2 d e\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e (5 b d-12 a e)+2 c d^2\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\left (-12 b \left (a e^2+c d^2\right )+28 a c d e+5 b^2 d e\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e (5 b d-12 a e)+2 c d^2\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\left (-12 b \left (a e^2+c d^2\right )+28 a c d e+5 b^2 d e\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e (5 b d-12 a e)+2 c d^2\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\left (-12 b \left (a e^2+c d^2\right )+28 a c d e+5 b^2 d e\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e (5 b d-12 a e)+2 c d^2\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

(d*(a + b*x + c*x^2)^(5/2))/(6*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) - (-1/ 
5*((2*c*d^2 + e*(5*b*d - 12*a*e))*(a + b*x + c*x^2)^(5/2))/((c*d^2 - b*d*e 
 + a*e^2)*(d + e*x)^5) - ((5*b^2*d*e + 28*a*c*d*e - 12*b*(c*d^2 + a*e^2))* 
(((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*d^2 - b*d 
*e + a*e^2)*(d + e*x)^4) - (3*(b^2 - 4*a*c)*(((b*d - 2*a*e + (2*c*d - b*e) 
*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((b^2 
 - 4*a*c)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + 
a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*e^2)^(3/2))))/(16*(c 
*d^2 - b*d*e + a*e^2))))/(2*(c*d^2 - b*d*e + a*e^2)))/(12*(c*d^2 - b*d*e + 
 a*e^2))
 

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 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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(15587\) vs. \(2(388)=776\).

Time = 2.30 (sec) , antiderivative size = 15588, normalized size of antiderivative = 37.65

method	result	size
default	\(\text {Expression too large to display}\)	\(15588\)

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4039 vs. \(2 (388) = 776\).

Time = 114.28 (sec) , antiderivative size = 8120, normalized size of antiderivative = 19.61 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^7,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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15909 vs. \(2 (388) = 776\).

Time = 13.01 (sec) , antiderivative size = 15909, normalized size of antiderivative = 38.43 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/512*(12*b^5*c*d^2 - 96*a*b^3*c^2*d^2 + 192*a^2*b*c^3*d^2 - 5*b^6*d*e + 
12*a*b^4*c*d*e + 144*a^2*b^2*c^2*d*e - 448*a^3*c^3*d*e + 12*a*b^5*e^2 - 96 
*a^2*b^3*c*e^2 + 192*a^3*b*c^2*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^ 
7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d 
^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^ 
3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b 
*d*e^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/7680*(180*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^11*b^5*c*d^2*e^10 - 1440*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^11*a*b^3*c^2*d^2*e^10 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^11*a^2*b*c^3*d^2*e^10 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^6*d* 
e^11 + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^4*c*d*e^11 + 2160*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b^2*c^2*d*e^11 - 6720*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^11*a^3*c^3*d*e^11 + 180*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^11*a*b^5*e^12 - 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2 
*b^3*c*e^12 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^3*b*c^2*e^12 + 
 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*c^(13/2)*d^8*e^4 - 61440*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^10*b*c^(11/2)*d^7*e^5 + 92160*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^10*b^2*c^(9/2)*d^6*e^6 + 61440*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^10*a*c^(11/2)*d^6*e^6 - 61440*(sqrt(c)*x - sqrt(c*x^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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