\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [592]

Optimal result

Integrand size = 22, antiderivative size = 409 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^{9/2}} \] Output:

-1/512*(-4*a*c+b^2)*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(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*(2 
4*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(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*e*(c*x^2+b*x+a)^(5/2)/(a*e 
^2-b*d*e+c*d^2)/(e*x+d)^6-7/60*e*(-b*e+2*c*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*(24*c^2*d^2+7*b^2*e^2-4*c*e* 
(a*e+6*b*d))*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 = 11.64 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {e (a+x (b+c x))^{5/2}}{6 \left (c d^2+e (-b d+a e)\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) (a+x (b+c x))^{5/2}}{60 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^5}-\frac {\left (12 c^2 d^2+\frac {7 b^2 e^2}{2}-2 c e (6 b d+a e)\right ) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{192 \left (c d^2+e (-b d+a e)\right )^3} \] Input:

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

Output:

-1/6*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^6) 
- (7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(60*(c*d^2 + e*(-(b*d) + a*e 
))^2*(d + e*x)^5) - ((12*c^2*d^2 + (7*b^2*e^2)/2 - 2*c*e*(6*b*d + a*e))*(( 
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)*((Sqrt[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)))))/(192*(c*d^2 + e*(- 
(b*d) + a*e))^3)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1167, 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.

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

\(\Big \downarrow \) 1167

\(\displaystyle -\frac {\int -\frac {(12 c d-7 b e-2 c e x) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)^6}dx}{6 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(12 c d-7 b e-2 c e x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^6}dx}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(d+e x)^5}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}}{12 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-1/6*(e*(a + b*x + c*x^2)^(5/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) + ( 
(-7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d 
 + e*x)^5) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(((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)*Arc 
Tanh[(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 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(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)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10396\) vs. \(2(383)=766\).

Time = 2.07 (sec) , antiderivative size = 10397, normalized size of antiderivative = 25.42

method	result	size
default	\(\text {Expression too large to display}\)	\(10397\)

Input:

int((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. 4008 vs. \(2 (383) = 766\).

Time = 108.81 (sec) , antiderivative size = 8058, normalized size of antiderivative = 19.70 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((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. 14486 vs. \(2 (383) = 766\).

Time = 10.50 (sec) , antiderivative size = 14486, normalized size of antiderivative = 35.42 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

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

Output:

1/512*(24*b^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e 
 + 192*a*b^3*c^2*d*e - 384*a^2*b*c^3*d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 14 
4*a^2*b^2*c^2*e^2 - 64*a^3*c^3*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*(360*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^11*b^4*c^2*d^2*e^9 - 2880*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^11*a*b^2*c^3*d^2*e^9 + 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^11*a^2*c^4*d^2*e^9 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c*d* 
e^10 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^2*d*e^10 - 5760 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b*c^3*d*e^10 + 105*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^11*b^6*e^11 - 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^11*a*b^4*c*e^11 + 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b^2 
*c^2*e^11 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^3*c^3*e^11 + 3960 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^4*c^(5/2)*d^3*e^8 - 31680*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^2*c^(7/2)*d^3*e^8 + 63360*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^10*a^2*c^(9/2)*d^3*e^8 - 3960*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^10*b^5*c^(3/2)*d^2*e^9 + 31680*(sqrt(c)*x - sqrt(c*x^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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