Integrand size = 25, antiderivative size = 297 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=-\frac {5 \left (b^2-4 a c\right ) \left (44 c^2 d f+13 b^2 e g-8 a c e g-22 b c (e f+d g)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{3696 c^4}+\frac {\left (44 c^2 d f+13 b^2 e g-8 a c e g-22 b c (e f+d g)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}-\frac {(13 b e g-22 c (e f+d g)-18 c e g x) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {5 \left (-b^2+4 a c\right )^{5/2} \left (44 c^2 d f+13 b^2 e g-8 a c e g-22 b c (e f+d g)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{3696 \sqrt {2} c^5 \left (a+b x+c x^2\right )^{3/4}} \] Output:
-5/3696*(-4*a*c+b^2)*(44*c^2*d*f+13*b^2*e*g-8*a*c*e*g-22*b*c*(d*g+e*f))*(2 *c*x+b)*(c*x^2+b*x+a)^(1/4)/c^4+1/308*(44*c^2*d*f+13*b^2*e*g-8*a*c*e*g-22* b*c*(d*g+e*f))*(2*c*x+b)*(c*x^2+b*x+a)^(5/4)/c^3-1/99*(13*b*e*g-22*c*(d*g+ e*f)-18*c*e*g*x)*(c*x^2+b*x+a)^(9/4)/c^2+5/7392*(4*a*c-b^2)^(5/2)*(44*c^2* d*f+13*b^2*e*g-8*a*c*e*g-22*b*c*(d*g+e*f))*(-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)
Time = 10.73 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.76 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {(a+x (b+c x))^{9/4} (-13 b e g+2 c (11 e f+11 d g+9 e g x))+\frac {3 \left (\frac {13}{4} b^2 e g-2 a c e g+\frac {11}{2} c (2 c d f-b (e f+d g))\right ) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )\right )}{56 c^3 (a+x (b+c x))^{3/4}}}{99 c^2} \] Input:
Integrate[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(5/4),x]
Output:
((a + x*(b + c*x))^(9/4)*(-13*b*e*g + 2*c*(11*e*f + 11*d*g + 9*e*g*x)) + ( 3*((13*b^2*e*g)/4 - 2*a*c*e*g + (11*c*(2*c*d*f - b*(e*f + d*g)))/2)*(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])))/(56 *c^3*(a + x*(b + c*x))^(3/4)))/(99*c^2)
Time = 0.76 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
| \(\displaystyle \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\left (-8 a c e g+13 b^2 e g+22 c (2 c d f-b (d g+e f))\right ) \int \left (c x^2+b x+a\right )^{5/4}dx}{44 c^2}-\frac {\left (a+b x+c x^2\right )^{9/4} (13 b e g-22 c (d g+e f)-18 c e g x)}{99 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\left (-8 a c e g+13 b^2 e g+22 c (2 c d f-b (d g+e f))\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c}\right )}{44 c^2}-\frac {\left (a+b x+c x^2\right )^{9/4} (13 b e g-22 c (d g+e f)-18 c e g x)}{99 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\left (-8 a c e g+13 b^2 e g+22 c (2 c d f-b (d g+e f))\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c}\right )}{44 c^2}-\frac {\left (a+b x+c x^2\right )^{9/4} (13 b e g-22 c (d g+e f)-18 c e g x)}{99 c^2}\) |
| \(\Big \downarrow \) 1094 |
| \(\displaystyle \frac {\left (-8 a c e g+13 b^2 e g+22 c (2 c d f-b (d g+e f))\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c}\right )}{44 c^2}-\frac {\left (a+b x+c x^2\right )^{9/4} (13 b e g-22 c (d g+e f)-18 c e g x)}{99 c^2}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{28 c}\right ) \left (-8 a c e g+13 b^2 e g+22 c (2 c d f-b (d g+e f))\right )}{44 c^2}-\frac {\left (a+b x+c x^2\right )^{9/4} (13 b e g-22 c (d g+e f)-18 c e g x)}{99 c^2}\) |
Input:
Int[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(5/4),x]
Output:
-1/99*((13*b*e*g - 22*c*(e*f + d*g) - 18*c*e*g*x)*(a + b*x + c*x^2)^(9/4)) /c^2 + ((13*b^2*e*g - 8*a*c*e*g + 22*c*(2*c*d*f - b*(e*f + d*g)))*(((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)*Sq rt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(28*c)))/(44*c^2)
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]
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])
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]
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]
\[\int \left (e x +d \right ) \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}d x\]
Input:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
Output:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
Output:
integral((c*e*g*x^4 + (c*e*f + (c*d + b*e)*g)*x^3 + a*d*f + ((c*d + b*e)*f + (b*d + a*e)*g)*x^2 + (a*d*g + (b*d + a*e)*f)*x)*(c*x^2 + b*x + a)^(1/4) , x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (d + e x\right ) \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \] Input:
integrate((e*x+d)*(g*x+f)*(c*x**2+b*x+a)**(5/4),x)
Output:
Integral((d + e*x)*(f + g*x)*(a + b*x + c*x**2)**(5/4), x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)*(g*x + f), x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)*(g*x + f), x)
Timed out. \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (f+g\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \] Input:
int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(5/4),x)
Output:
int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(5/4), x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx =\text {Too large to display} \] Input:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
Output:
( - 3840*(a + b*x + c*x**2)**(1/4)*a**3*c**2*e*g + 3712*(a + b*x + c*x**2) **(1/4)*a**2*b**2*c*e*g - 5632*(a + b*x + c*x**2)**(1/4)*a**2*b*c**2*d*g - 5632*(a + b*x + c*x**2)**(1/4)*a**2*b*c**2*e*f + 960*(a + b*x + c*x**2)** (1/4)*a**2*b*c**2*e*g*x + 21120*(a + b*x + c*x**2)**(1/4)*a**2*c**3*d*f - 624*(a + b*x + c*x**2)**(1/4)*a*b**4*e*g + 1056*(a + b*x + c*x**2)**(1/4)* a*b**3*c*d*g + 1056*(a + b*x + c*x**2)**(1/4)*a*b**3*c*e*f - 928*(a + b*x + c*x**2)**(1/4)*a*b**3*c*e*g*x - 2112*(a + b*x + c*x**2)**(1/4)*a*b**2*c* *2*d*f + 1408*(a + b*x + c*x**2)**(1/4)*a*b**2*c**2*d*g*x + 1408*(a + b*x + c*x**2)**(1/4)*a*b**2*c**2*e*f*x + 512*(a + b*x + c*x**2)**(1/4)*a*b**2* c**2*e*g*x**2 + 16896*(a + b*x + c*x**2)**(1/4)*a*b*c**3*d*f*x + 9856*(a + b*x + c*x**2)**(1/4)*a*b*c**3*d*g*x**2 + 9856*(a + b*x + c*x**2)**(1/4)*a *b*c**3*e*f*x**2 + 6912*(a + b*x + c*x**2)**(1/4)*a*b*c**3*e*g*x**3 + 156* (a + b*x + c*x**2)**(1/4)*b**5*e*g*x - 264*(a + b*x + c*x**2)**(1/4)*b**4* c*d*g*x - 264*(a + b*x + c*x**2)**(1/4)*b**4*c*e*f*x - 104*(a + b*x + c*x* *2)**(1/4)*b**4*c*e*g*x**2 + 528*(a + b*x + c*x**2)**(1/4)*b**3*c**2*d*f*x + 176*(a + b*x + c*x**2)**(1/4)*b**3*c**2*d*g*x**2 + 176*(a + b*x + c*x** 2)**(1/4)*b**3*c**2*e*f*x**2 + 80*(a + b*x + c*x**2)**(1/4)*b**3*c**2*e*g* x**3 + 9504*(a + b*x + c*x**2)**(1/4)*b**2*c**3*d*f*x**2 + 6688*(a + b*x + c*x**2)**(1/4)*b**2*c**3*d*g*x**3 + 6688*(a + b*x + c*x**2)**(1/4)*b**2*c **3*e*f*x**3 + 5152*(a + b*x + c*x**2)**(1/4)*b**2*c**3*e*g*x**4 + 6336...