Integrand size = 27, antiderivative size = 378 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {\left (72 c^3 d^2 f-15 b^3 e^2 g+2 b c e (12 a e g+11 b (e f+2 d g))-4 c^2 (9 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{360 c^4}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {\left (165 b^2 e^2 g+72 c^2 d (11 e f+2 d g)-2 c e (72 a e g+121 b (e f+2 d g))+14 c e (22 c e f+8 c d g-15 b e g) x\right ) \left (a+b x+c x^2\right )^{7/4}}{1386 c^3}-\frac {\left (b^2-4 a c\right )^{3/2} \left (72 c^3 d^2 f-15 b^3 e^2 g+2 b c e (12 a e g+11 b (e f+2 d g))-4 c^2 (9 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{240 \sqrt {2} c^5 \sqrt [4]{a+b x+c x^2}} \] Output:
1/360*(72*c^3*d^2*f-15*b^3*e^2*g+2*b*c*e*(12*a*e*g+11*b*(2*d*g+e*f))-4*c^2 *(9*b*d*(d*g+2*e*f)+4*a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/4)/c^4+ 2/11*g*(e*x+d)^2*(c*x^2+b*x+a)^(7/4)/c+1/1386*(165*b^2*e^2*g+72*c^2*d*(2*d *g+11*e*f)-2*c*e*(72*a*e*g+121*b*(2*d*g+e*f))+14*c*e*(-15*b*e*g+8*c*d*g+22 *c*e*f)*x)*(c*x^2+b*x+a)^(7/4)/c^3-1/480*(-4*a*c+b^2)^(3/2)*(72*c^3*d^2*f- 15*b^3*e^2*g+2*b*c*e*(12*a*e*g+11*b*(2*d*g+e*f))-4*c^2*(9*b*d*(d*g+2*e*f)+ 4*a*e*(2*d*g+e*f)))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/ 2*arcsin((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) ^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.90 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.77 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {40320 c^4 g (d+e x)^2 (a+x (b+c x))^2+160 c^2 (a+x (b+c x))^2 \left (165 b^2 e^2 g+4 c^2 \left (36 d^2 g+77 e^2 f x+2 d e (99 f+14 g x)\right )-2 c e (72 a e g+b (121 e f+242 d g+105 e g x))\right )+77 \left (72 c^3 d^2 f-15 b^3 e^2 g+2 b c e (12 a e g+11 b (e f+2 d g))-4 c^2 (9 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) (b+2 c x) \left (8 c (a+x (b+c x))-3 \sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{221760 c^5 \sqrt [4]{a+x (b+c x)}} \] Input:
Integrate[(d + e*x)^2*(f + g*x)*(a + b*x + c*x^2)^(3/4),x]
Output:
(40320*c^4*g*(d + e*x)^2*(a + x*(b + c*x))^2 + 160*c^2*(a + x*(b + c*x))^2 *(165*b^2*e^2*g + 4*c^2*(36*d^2*g + 77*e^2*f*x + 2*d*e*(99*f + 14*g*x)) - 2*c*e*(72*a*e*g + b*(121*e*f + 242*d*g + 105*e*g*x))) + 77*(72*c^3*d^2*f - 15*b^3*e^2*g + 2*b*c*e*(12*a*e*g + 11*b*(e*f + 2*d*g)) - 4*c^2*(9*b*d*(2* e*f + d*g) + 4*a*e*(e*f + 2*d*g)))*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3* Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hyperge ometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(221760*c^5*(a + x*(b + c*x))^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(781\) vs. \(2(378)=756\).
Time = 1.38 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1225, 1087, 1094, 834, 761, 1510}
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.
| \(\displaystyle \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 \int \frac {1}{4} (d+e x) (22 c d f-7 b d g-8 a e g+(22 c e f+8 c d g-15 b e g) x) \left (c x^2+b x+a\right )^{3/4}dx}{11 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d+e x) (22 c d f-7 b d g-8 a e g+(22 c e f+8 c d g-15 b e g) x) \left (c x^2+b x+a\right )^{3/4}dx}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {11 \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right ) \int \left (c x^2+b x+a\right )^{3/4}dx}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {11 \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{20 c}\right )}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 1094 |
| \(\displaystyle \frac {\frac {11 \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{5 c (b+2 c x)}\right )}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {11 \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right )}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {11 \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right )}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {11 \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt [4]{a+b x+c x^2} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right ) \left (-4 c^2 (4 a e (2 d g+e f)+9 b d (d g+2 e f))+2 b c e (12 a e g+11 b (2 d g+e f))-15 b^3 e^2 g+72 c^3 d^2 f\right )}{36 c^2}+\frac {\left (a+b x+c x^2\right )^{7/4} \left (-2 c e (72 a e g+121 b (2 d g+e f))+165 b^2 e^2 g+14 c e x (-15 b e g+8 c d g+22 c e f)+72 c^2 d (2 d g+11 e f)\right )}{63 c^2}}{22 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}\) |
Input:
Int[(d + e*x)^2*(f + g*x)*(a + b*x + c*x^2)^(3/4),x]
Output:
(2*g*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c) + (((165*b^2*e^2*g + 72*c ^2*d*(11*e*f + 2*d*g) - 2*c*e*(72*a*e*g + 121*b*(e*f + 2*d*g)) + 14*c*e*(2 2*c*e*f + 8*c*d*g - 15*b*e*g)*x)*(a + b*x + c*x^2)^(7/4))/(63*c^2) + (11*( 72*c^3*d^2*f - 15*b^3*e^2*g + 2*b*c*e*(12*a*e*g + 11*b*(e*f + 2*d*g)) - 4* c^2*(9*b*d*(2*e*f + d*g) + 4*a*e*(e*f + 2*d*g)))*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(5*c) - (3*(b^2 - 4*a*c)*Sqrt[(b + 2*c*x)^2]*(-1/2*(Sqrt[b^2 - 4*a*c]*(-(((a + b*x + c*x^2)^(1/4)*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]))) + ((b^2 - 4*a*c)^(1/4)*(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)]*EllipticE[2*A rcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2] )/(Sqrt[2]*c^(1/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/Sqrt[c] + ((b^2 - 4*a*c)^(3/4)*(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*Sqr t[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*EllipticF[2*ArcTan[(Sqr t[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(4*Sqrt[ 2]*c^(3/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(5*c*(b + 2*c*x))) )/(36*c^2))/(22*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
\[\int \left (e x +d \right )^{2} \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}d x\]
Input:
int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
Output:
int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="fricas")
Output:
integral((e^2*g*x^3 + d^2*f + (e^2*f + 2*d*e*g)*x^2 + (2*d*e*f + d^2*g)*x) *(c*x^2 + b*x + a)^(3/4), x)
\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int \left (d + e x\right )^{2} \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{4}}\, dx \] Input:
integrate((e*x+d)**2*(g*x+f)*(c*x**2+b*x+a)**(3/4),x)
Output:
Integral((d + e*x)**2*(f + g*x)*(a + b*x + c*x**2)**(3/4), x)
\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d)^2*(g*x + f), x)
\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d)^2*(g*x + f), x)
Timed out. \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/4} \,d x \] Input:
int((f + g*x)*(d + e*x)^2*(a + b*x + c*x^2)^(3/4),x)
Output:
int((f + g*x)*(d + e*x)^2*(a + b*x + c*x^2)^(3/4), x)
\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
Output:
(9024*(a + b*x + c*x**2)**(3/4)*a**2*b*c*e**2*g - 19712*(a + b*x + c*x**2) **(3/4)*a**2*c**2*d*e*g - 9856*(a + b*x + c*x**2)**(3/4)*a**2*c**2*e**2*f - 2640*(a + b*x + c*x**2)**(3/4)*a*b**3*e**2*g + 7744*(a + b*x + c*x**2)** (3/4)*a*b**2*c*d*e*g + 3872*(a + b*x + c*x**2)**(3/4)*a*b**2*c*e**2*f - 67 68*(a + b*x + c*x**2)**(3/4)*a*b**2*c*e**2*g*x - 6336*(a + b*x + c*x**2)** (3/4)*a*b*c**2*d**2*g - 12672*(a + b*x + c*x**2)**(3/4)*a*b*c**2*d*e*f + 1 4784*(a + b*x + c*x**2)**(3/4)*a*b*c**2*d*e*g*x + 7392*(a + b*x + c*x**2)* *(3/4)*a*b*c**2*e**2*f*x + 4320*(a + b*x + c*x**2)**(3/4)*a*b*c**2*e**2*g* x**2 + 44352*(a + b*x + c*x**2)**(3/4)*a*c**3*d**2*f + 1980*(a + b*x + c*x **2)**(3/4)*b**4*e**2*g*x - 5808*(a + b*x + c*x**2)**(3/4)*b**3*c*d*e*g*x - 2904*(a + b*x + c*x**2)**(3/4)*b**3*c*e**2*f*x - 1800*(a + b*x + c*x**2) **(3/4)*b**3*c*e**2*g*x**2 + 4752*(a + b*x + c*x**2)**(3/4)*b**2*c**2*d**2 *g*x + 9504*(a + b*x + c*x**2)**(3/4)*b**2*c**2*d*e*f*x + 5280*(a + b*x + c*x**2)**(3/4)*b**2*c**2*d*e*g*x**2 + 2640*(a + b*x + c*x**2)**(3/4)*b**2* c**2*e**2*f*x**2 + 1680*(a + b*x + c*x**2)**(3/4)*b**2*c**2*e**2*g*x**3 + 22176*(a + b*x + c*x**2)**(3/4)*b*c**3*d**2*f*x + 15840*(a + b*x + c*x**2) **(3/4)*b*c**3*d**2*g*x**2 + 31680*(a + b*x + c*x**2)**(3/4)*b*c**3*d*e*f* x**2 + 24640*(a + b*x + c*x**2)**(3/4)*b*c**3*d*e*g*x**3 + 12320*(a + b*x + c*x**2)**(3/4)*b*c**3*e**2*f*x**3 + 10080*(a + b*x + c*x**2)**(3/4)*b*c* *3*e**2*g*x**4 - 22176*int(((a + b*x + c*x**2)**(3/4)*x)/(a + b*x + c*x...