\(\int (d+e x)^3 (a+b x+c x^2)^{5/4} \, dx\) [734]

Optimal result

Integrand size = 22, antiderivative size = 370 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{7392 c^5}+\frac {(2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac {e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac {5 \left (-b^2+4 a c\right )^{5/2} (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{7392 \sqrt {2} c^6 \left (a+b x+c x^2\right )^{3/4}} \] Output:

-5/7392*(-4*a*c+b^2)*(-b*e+2*c*d)*(44*c^2*d^2+17*b^2*e^2-4*c*e*(6*a*e+11*b 
*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)/c^5+1/616*(-b*e+2*c*d)*(44*c^2*d^2+17*b 
^2*e^2-4*c*e*(6*a*e+11*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(5/4)/c^4+2/13*e*(e*x 
+d)^2*(c*x^2+b*x+a)^(9/4)/c+1/2574*e*(1320*c^2*d^2+221*b^2*e^2-2*c*e*(88*a 
*e+507*b*d)+306*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(9/4)/c^3+5/14784*(4*a*c 
-b^2)^(5/2)*(-b*e+2*c*d)*(44*c^2*d^2+17*b^2*e^2-4*c*e*(6*a*e+11*b*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^6/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 10.85 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.77 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {88704 c^5 e (d+e x)^2 (a+x (b+c x))^3+224 c^3 e (a+x (b+c x))^3 \left (221 b^2 e^2+12 c^2 d (110 d+51 e x)-2 c e (507 b d+88 a e+153 b e x)\right )+39 (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) \left (2 c (b+2 c x) \left (32 a^2 c+a \left (-5 b^2+44 b c x+44 c^2 x^2\right )+x \left (-5 b^3+7 b^2 c x+24 b c^2 x^2+12 c^3 x^3\right )\right )+5 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )}{576576 c^6 (a+x (b+c x))^{3/4}} \] Input:

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(88704*c^5*e*(d + e*x)^2*(a + x*(b + c*x))^3 + 224*c^3*e*(a + x*(b + c*x)) 
^3*(221*b^2*e^2 + 12*c^2*d*(110*d + 51*e*x) - 2*c*e*(507*b*d + 88*a*e + 15 
3*b*e*x)) + 39*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*e^2 - 4*c*e*(11*b*d + 6* 
a*e))*(2*c*(b + 2*c*x)*(32*a^2*c + a*(-5*b^2 + 44*b*c*x + 44*c^2*x^2) + x* 
(-5*b^3 + 7*b^2*c*x + 24*b*c^2*x^2 + 12*c^3*x^3)) + 5*Sqrt[2]*(b^2 - 4*a*c 
)^(5/2)*((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]))/(576576*c^6*(a + x*(b + c*x))^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.53 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1166, 27, 1225, 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.

\(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {1}{4} (d+e x) \left (26 c d^2-9 b e d-8 a e^2+17 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/4}dx}{13 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) \left (26 c d^2-9 b e d-8 a e^2+17 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/4}dx}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {13 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \int \left (c x^2+b x+a\right )^{5/4}dx}{44 c^2}+\frac {e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{99 c^2}}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {13 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c}\right )}{44 c^2}+\frac {e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{99 c^2}}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {13 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c}\right )}{44 c^2}+\frac {e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{99 c^2}}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {13 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c}\right )}{44 c^2}+\frac {e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{99 c^2}}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {13 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{28 c}\right )}{44 c^2}+\frac {e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{99 c^2}}{26 c}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

Input:

Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(9/4))/(13*c) + ((e*(1320*c^2*d^2 + 221 
*b^2*e^2 - 2*c*e*(507*b*d + 88*a*e) + 306*c*e*(2*c*d - b*e)*x)*(a + b*x + 
c*x^2)^(9/4))/(99*c^2) + (13*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*e^2 - 4*c* 
e*(11*b*d + 6*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*Sq 
rt[2]*c^(5/4)*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(28 
*c)))/(44*c^2))/(26*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 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]
 

rule 1225

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

Maple [F]

\[\int \left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}d x\]

Input:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x)
 

Output:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x)
 

Fricas [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((c*e^3*x^5 + (3*c*d*e^2 + b*e^3)*x^4 + a*d^3 + (3*c*d^2*e + 3*b*d 
*e^2 + a*e^3)*x^3 + (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^2 + (b*d^3 + 3*a*d^2 
*e)*x)*(c*x^2 + b*x + a)^(1/4), x)
 

Sympy [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \] Input:

integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/4),x)
 

Output:

Integral((d + e*x)**3*(a + b*x + c*x**2)**(5/4), x)
 

Maxima [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3, x)
 

Giac [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \] Input:

int((d + e*x)^3*(a + b*x + c*x^2)^(5/4),x)
 

Output:

int((d + e*x)^3*(a + b*x + c*x^2)^(5/4), x)
 

Reduce [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx=\text {too large to display} \] Input:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x)
                                                                                    
                                                                                    
 

Output:

(110336*(a + b*x + c*x**2)**(1/4)*a**3*b*c**2*e**3 - 299520*(a + b*x + c*x 
**2)**(1/4)*a**3*c**3*d*e**2 - 71552*(a + b*x + c*x**2)**(1/4)*a**2*b**3*c 
*e**3 + 289536*(a + b*x + c*x**2)**(1/4)*a**2*b**2*c**2*d*e**2 - 27584*(a 
+ b*x + c*x**2)**(1/4)*a**2*b**2*c**2*e**3*x - 439296*(a + b*x + c*x**2)** 
(1/4)*a**2*b*c**3*d**2*e + 74880*(a + b*x + c*x**2)**(1/4)*a**2*b*c**3*d*e 
**2*x + 9856*(a + b*x + c*x**2)**(1/4)*a**2*b*c**3*e**3*x**2 + 549120*(a + 
 b*x + c*x**2)**(1/4)*a**2*c**4*d**3 + 10608*(a + b*x + c*x**2)**(1/4)*a*b 
**5*e**3 - 48672*(a + b*x + c*x**2)**(1/4)*a*b**4*c*d*e**2 + 17888*(a + b* 
x + c*x**2)**(1/4)*a*b**4*c*e**3*x + 82368*(a + b*x + c*x**2)**(1/4)*a*b** 
3*c**2*d**2*e - 72384*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*d*e**2*x - 101 
12*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*e**3*x**2 - 54912*(a + b*x + c*x* 
*2)**(1/4)*a*b**2*c**3*d**3 + 109824*(a + b*x + c*x**2)**(1/4)*a*b**2*c**3 
*d**2*e*x + 39936*(a + b*x + c*x**2)**(1/4)*a*b**2*c**3*d*e**2*x**2 + 6400 
*(a + b*x + c*x**2)**(1/4)*a*b**2*c**3*e**3*x**3 + 439296*(a + b*x + c*x** 
2)**(1/4)*a*b*c**4*d**3*x + 768768*(a + b*x + c*x**2)**(1/4)*a*b*c**4*d**2 
*e*x**2 + 539136*(a + b*x + c*x**2)**(1/4)*a*b*c**4*d*e**2*x**3 + 137984*( 
a + b*x + c*x**2)**(1/4)*a*b*c**4*e**3*x**4 - 2652*(a + b*x + c*x**2)**(1/ 
4)*b**6*e**3*x + 12168*(a + b*x + c*x**2)**(1/4)*b**5*c*d*e**2*x + 1768*(a 
 + b*x + c*x**2)**(1/4)*b**5*c*e**3*x**2 - 20592*(a + b*x + c*x**2)**(1/4) 
*b**4*c**2*d**2*e*x - 8112*(a + b*x + c*x**2)**(1/4)*b**4*c**2*d*e**2*x...