∫ (d + ex)3∕2(a + bx + cx2)3∕2 dx [2447]

Optimal. Leaf size=816 \[ \frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-21 a e)+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b d e-20 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

2/231*(c^2*d^2-6*b^2*e^2+c*e*(-3*a*e+13*b*d)+14*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)*(e*x+d)^(1/2)/c^2/e+2/

11*e*(c*x^2+b*x+a)^(5/2)*(e*x+d)^(1/2)/c+2/1155*(8*c^4*d^4+8*b^4*e^4-c^3*d^2*e*(-42*a*e+19*b*d)-b^2*c*e^3*(21*

a*e+19*b*d)+3*c^2*e^2*(-10*a^2*e^2+17*a*b*d*e+2*b^2*d^2)-3*c*e*(-b*e+2*c*d)*(c^2*d^2+8*b^2*e^2-c*e*(31*a*e+b*d

))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/e^3-8/1155*(-b*e+2*c*d)*(c^2*d^2-2*b^2*e^2-c*e*(-9*a*e+b*d))*(c^2*

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

*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c

*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/e^4/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)

+2/1155*(a*e^2-b*d*e+c*d^2)*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(-21*a*e+8*b*d)+b^2*c*e^3*(51*a*e+13*b*d)+3*c^2*

e^2*(-20*a^2*e^2-28*a*b*d*e+b^2*d^2))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^

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

b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^4/e^4/(e*x+d)^(1/2)/(c*x^2+b*x

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Rubi [A]
time = 1.87, antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {756, 828, 857, 732, 435, 430} \begin {gather*} \frac {2 e \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{1155 c^3 e^3}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {c x^2+b x+a}} \end {gather*}

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

(2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^3*(19*b*d + 21*a*e) + 3*c^2*e^

2*(2*b^2*d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sq

rt[a + b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) + 14*c*e*(2

*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (8*

Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*

d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4

*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(11

55*c^4*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^

2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 21*a*e) + b^2*c*e^3*(13*b*d

+ 51*a*e) + 3*c^2*e^2*(b^2*d^2 - 28*a*b*d*e - 20*a^2*e^2))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])

*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b

^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(1155*c^4*e^4*Sqrt[d + e

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]

))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt

Q[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell

ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*

(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*

e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2

])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b

, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*

((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim

p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^

2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a

+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -

c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*

d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0

] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])

) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis

t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^

2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (11 c d^2-e (5 b d+a e)\right )+3 e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 c}\\ &=\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {4 \int \frac {\left (\frac {3}{4} e \left (b c^2 d^3+13 b^2 c d^2 e-58 a c^2 d^2 e-6 b^3 d e^2+27 a b c d e^2-2 a b^2 e^3+6 a^2 c e^3\right )+\frac {3}{4} e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {-\frac {3}{8} e \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )-3 e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac {\left (8 \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac {3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e^5}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (16 \sqrt {2} \sqrt {b^2-4 a c} \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac {3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3465 c^4 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+84 a c^3 d^2 e^2+13 b^3 c d e^3-84 a b c^2 d e^3-8 b^4 e^4+51 a b^2 c e^4-60 a^2 c^2 e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 32.96, size = 10848, normalized size = 13.29 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11932\) vs. \(2(746)=1492\).
time = 1.08, size = 11933, normalized size = 14.62


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(2888\)
risch	\(\text {Expression too large to display}\)	\(4982\)
default	\(\text {Expression too large to display}\)	\(11933\)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.55, size = 869, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 3 \, {\left (11 \, b^{2} c^{4} + 36 \, a c^{5}\right )} d^{4} e^{2} + 2 \, {\left (7 \, b^{3} c^{3} - 108 \, a b c^{4}\right )} d^{3} e^{3} + 3 \, {\left (11 \, b^{4} c^{2} - 102 \, a b^{2} c^{3} + 312 \, a^{2} c^{4}\right )} d^{2} e^{4} - 6 \, {\left (8 \, b^{5} c - 69 \, a b^{3} c^{2} + 156 \, a^{2} b c^{3}\right )} d e^{5} + {\left (16 \, b^{6} - 144 \, a b^{4} c + 369 \, a^{2} b^{2} c^{2} - 180 \, a^{3} c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, {\left (b^{2} c^{4} + 6 \, a c^{5}\right )} d^{3} e^{3} + 2 \, {\left (b^{3} c^{3} - 9 \, a b c^{4}\right )} d^{2} e^{4} - {\left (5 \, b^{4} c^{2} - 36 \, a b^{2} c^{3} + 54 \, a^{2} c^{4}\right )} d e^{5} + {\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 27 \, a^{2} b c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{6} d^{4} e^{2} + {\left (105 \, c^{6} x^{4} + 140 \, b c^{5} x^{3} + 8 \, b^{4} c^{2} - 51 \, a b^{2} c^{3} + 60 \, a^{2} c^{4} + 5 \, {\left (b^{2} c^{4} + 39 \, a c^{5}\right )} x^{2} - 2 \, {\left (3 \, b^{3} c^{3} - 16 \, a b c^{4}\right )} x\right )} e^{6} + {\left (140 \, c^{6} d x^{3} + 205 \, b c^{5} d x^{2} + 2 \, {\left (7 \, b^{2} c^{4} + 163 \, a c^{5}\right )} d x - {\left (19 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} d\right )} e^{5} + {\left (5 \, c^{6} d^{2} x^{2} + 14 \, b c^{5} d^{2} x + {\left (6 \, b^{2} c^{4} + 47 \, a c^{5}\right )} d^{2}\right )} e^{4} - {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3465 \, c^{5}} \end {gather*}

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

2/3465*((16*c^6*d^6 - 48*b*c^5*d^5*e + 3*(11*b^2*c^4 + 36*a*c^5)*d^4*e^2 + 2*(7*b^3*c^3 - 108*a*b*c^4)*d^3*e^3

+ 3*(11*b^4*c^2 - 102*a*b^2*c^3 + 312*a^2*c^4)*d^2*e^4 - 6*(8*b^5*c - 69*a*b^3*c^2 + 156*a^2*b*c^3)*d*e^5 + (

16*b^6 - 144*a*b^4*c + 369*a^2*b^2*c^2 - 180*a^3*c^3)*e^6)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 -

b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3

- 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) + 24*(2*c^6*d^5*e - 5*b*c^5*d^4*e^2 + 2*(b^2*

c^4 + 6*a*c^5)*d^3*e^3 + 2*(b^3*c^3 - 9*a*b*c^4)*d^2*e^4 - (5*b^4*c^2 - 36*a*b^2*c^3 + 54*a^2*c^4)*d*e^5 + (2*

b^5*c - 15*a*b^3*c^2 + 27*a^2*b*c^3)*e^6)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*

c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-

3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2

*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) +

3*(8*c^6*d^4*e^2 + (105*c^6*x^4 + 140*b*c^5*x^3 + 8*b^4*c^2 - 51*a*b^2*c^3 + 60*a^2*c^4 + 5*(b^2*c^4 + 39*a*c

^5)*x^2 - 2*(3*b^3*c^3 - 16*a*b*c^4)*x)*e^6 + (140*c^6*d*x^3 + 205*b*c^5*d*x^2 + 2*(7*b^2*c^4 + 163*a*c^5)*d*x

- (19*b^3*c^3 - 116*a*b*c^4)*d)*e^5 + (5*c^6*d^2*x^2 + 14*b*c^5*d^2*x + (6*b^2*c^4 + 47*a*c^5)*d^2)*e^4 - (6*

c^6*d^3*x + 19*b*c^5*d^3)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))*e^(-5)/c^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}

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

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

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3.25.47 \(\int (d+e x)^{3/2} (a+b x+c x^2)^{3/2} \, dx\) [2447]