\(\int (d+e x)^2 (a+b x+c x^2)^{5/4} \, dx\) [735]

Optimal result

Integrand size = 22, antiderivative size = 275 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=-\frac {5 \left (b^2-4 a c\right ) \left (\left (13 b^2-8 a c\right ) e^2+44 c d (c d-b e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{3696 c^4}+\frac {\left (\left (13 b^2-8 a c\right ) e^2+44 c d (c d-b e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}+\frac {e (44 c d-13 b e+18 c e x) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {5 \left (-b^2+4 a c\right )^{5/2} \left (\left (13 b^2-8 a c\right ) e^2+44 c d (c d-b e)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{3696 \sqrt {2} c^5 \left (a+b x+c x^2\right )^{3/4}} \] Output:

-5/3696*(-4*a*c+b^2)*((-8*a*c+13*b^2)*e^2+44*c*d*(-b*e+c*d))*(2*c*x+b)*(c* 
x^2+b*x+a)^(1/4)/c^4+1/308*((-8*a*c+13*b^2)*e^2+44*c*d*(-b*e+c*d))*(2*c*x+ 
b)*(c*x^2+b*x+a)^(5/4)/c^3+1/99*e*(18*c*e*x-13*b*e+44*c*d)*(c*x^2+b*x+a)^( 
9/4)/c^2+5/7392*(4*a*c-b^2)^(5/2)*((-8*a*c+13*b^2)*e^2+44*c*d*(-b*e+c*d))* 
(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*InverseJacobiAM(1/2*arctan((2*c*x+b) 
/(4*a*c-b^2)^(1/2)),2^(1/2))*2^(1/2)/c^5/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 10.48 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {2 \left (\frac {13 e (2 c d-b e) (a+x (b+c x))^{9/4}}{18 c}+e (d+e x) (a+x (b+c x))^{9/4}-\frac {\left (-11 c^2 d^2-\frac {13 b^2 e^2}{4}+c e (11 b d+2 a e)\right ) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}}\right )}{11 c} \] Input:

Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(2*((13*e*(2*c*d - b*e)*(a + x*(b + c*x))^(9/4))/(18*c) + e*(d + e*x)*(a + 
 x*(b + c*x))^(9/4) - ((-11*c^2*d^2 - (13*b^2*e^2)/4 + c*e*(11*b*d + 2*a*e 
))*(24*c^2*(b + 2*c*x)*(a + x*(b + c*x))^2 - 5*(b^2 - 4*a*c)*(2*c*(b + 2*c 
*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x))) 
/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 
2])))/(336*c^4*(a + x*(b + c*x))^(3/4))))/(11*c)
 

Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.45, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1166, 27, 1160, 1087, 1087, 1094, 761}

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.

\(\displaystyle \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {1}{4} \left (22 c d^2-9 b e d-4 a e^2+13 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/4}dx}{11 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (22 c d^2-9 b e d-4 a e^2+13 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/4}dx}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \int \left (c x^2+b x+a\right )^{5/4}dx}{2 c}+\frac {26 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{9 c}}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c}\right )}{2 c}+\frac {26 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{9 c}}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c}\right )}{2 c}+\frac {26 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{9 c}}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c}\right )}{2 c}+\frac {26 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{9 c}}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {\left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{28 c}\right )}{2 c}+\frac {26 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{9 c}}{22 c}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}\)

Input:

Int[(d + e*x)^2*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(2*e*(d + e*x)*(a + b*x + c*x^2)^(9/4))/(11*c) + ((26*e*(2*c*d - b*e)*(a + 
 b*x + c*x^2)^(9/4))/(9*c) + ((44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d + 2 
*a*e))*(((b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(7*c) - (5*(b^2 - 4*a*c)*((( 
b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(3*c) - ((b^2 - 4*a*c)^(5/4)*Sqrt[(b + 
 2*c*x)^2]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[ 
(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[ 
a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/ 
4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(6*Sqrt[2]*c^(5/4) 
*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(28*c)))/(2*c))/ 
(22*c)
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1166

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)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

Maple [F]

\[\int \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}d x\]

Input:

int((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x)
 

Output:

int((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x)
 

Fricas [F]

\[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((c*e^2*x^4 + (2*c*d*e + b*e^2)*x^3 + a*d^2 + (c*d^2 + 2*b*d*e + a 
*e^2)*x^2 + (b*d^2 + 2*a*d*e)*x)*(c*x^2 + b*x + a)^(1/4), x)
 

Sympy [F]

\[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \] Input:

integrate((e*x+d)**2*(c*x**2+b*x+a)**(5/4),x)
 

Output:

Integral((d + e*x)**2*(a + b*x + c*x**2)**(5/4), x)
 

Maxima [F]

\[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2, x)
 

Giac [F]

\[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \] Input:

int((d + e*x)^2*(a + b*x + c*x^2)^(5/4),x)
 

Output:

int((d + e*x)^2*(a + b*x + c*x^2)^(5/4), x)
 

Reduce [F]

\[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2*(c*x^2+b*x+a)^(5/4),x)
                                                                                    
                                                                                    
 

Output:

( - 3840*(a + b*x + c*x**2)**(1/4)*a**3*c**2*e**2 + 3712*(a + b*x + c*x**2 
)**(1/4)*a**2*b**2*c*e**2 - 11264*(a + b*x + c*x**2)**(1/4)*a**2*b*c**2*d* 
e + 960*(a + b*x + c*x**2)**(1/4)*a**2*b*c**2*e**2*x + 21120*(a + b*x + c* 
x**2)**(1/4)*a**2*c**3*d**2 - 624*(a + b*x + c*x**2)**(1/4)*a*b**4*e**2 + 
2112*(a + b*x + c*x**2)**(1/4)*a*b**3*c*d*e - 928*(a + b*x + c*x**2)**(1/4 
)*a*b**3*c*e**2*x - 2112*(a + b*x + c*x**2)**(1/4)*a*b**2*c**2*d**2 + 2816 
*(a + b*x + c*x**2)**(1/4)*a*b**2*c**2*d*e*x + 512*(a + b*x + c*x**2)**(1/ 
4)*a*b**2*c**2*e**2*x**2 + 16896*(a + b*x + c*x**2)**(1/4)*a*b*c**3*d**2*x 
 + 19712*(a + b*x + c*x**2)**(1/4)*a*b*c**3*d*e*x**2 + 6912*(a + b*x + c*x 
**2)**(1/4)*a*b*c**3*e**2*x**3 + 156*(a + b*x + c*x**2)**(1/4)*b**5*e**2*x 
 - 528*(a + b*x + c*x**2)**(1/4)*b**4*c*d*e*x - 104*(a + b*x + c*x**2)**(1 
/4)*b**4*c*e**2*x**2 + 528*(a + b*x + c*x**2)**(1/4)*b**3*c**2*d**2*x + 35 
2*(a + b*x + c*x**2)**(1/4)*b**3*c**2*d*e*x**2 + 80*(a + b*x + c*x**2)**(1 
/4)*b**3*c**2*e**2*x**3 + 9504*(a + b*x + c*x**2)**(1/4)*b**2*c**3*d**2*x* 
*2 + 13376*(a + b*x + c*x**2)**(1/4)*b**2*c**3*d*e*x**3 + 5152*(a + b*x + 
c*x**2)**(1/4)*b**2*c**3*e**2*x**4 + 6336*(a + b*x + c*x**2)**(1/4)*b*c**4 
*d**2*x**3 + 9856*(a + b*x + c*x**2)**(1/4)*b*c**4*d*e*x**4 + 4032*(a + b* 
x + c*x**2)**(1/4)*b*c**4*e**2*x**5 + 1920*int(((a + b*x + c*x**2)**(1/4)* 
x)/(a + b*x + c*x**2),x)*a**3*c**3*e**2 - 4080*int(((a + b*x + c*x**2)**(1 
/4)*x)/(a + b*x + c*x**2),x)*a**2*b**2*c**2*e**2 + 10560*int(((a + b*x ...