3.25.0 ∫ (a+bx+cx2)5∕2 ______________________

Integrand size = 24, antiderivative size = 622 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

-2/3*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(3/2)+10/21*(2*c*e*x-7*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)/e^3/(e*x+d)^(1/2)+2/

21*(128*c^2*d^2+51*b^2*e^2-4*c*e*(-5*a*e+44*b*d)-48*c*e*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/e^5-

1/21*(-b*e+2*c*d)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(-29*a*e+32*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/e^6/(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)+4/21*(a*e^2-b*d*e+c*d^2)*(128*c^2*d^2+27*b^2*e^2-4*c*e*(-5*a*e+32*b*d))*Ell

ipticF(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/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (veriﬁed)

Time = 0.55 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {746, 826, 828, 857, 732, 435, 430} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{21 e^5}+\frac {10 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

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

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c

*x^2])/(21*e^5) + (10*(16*c*d - 7*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(21*e^3*Sqrt[d + e*x]) - (2*(a + b*x

+ c*x^2)^(5/2))/(3*e*(d + e*x)^(3/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4

*c*e*(32*b*d - 29*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)])/(21*c*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]

*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*a*e))*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)])/(21*c*e^6*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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[(d + e*x)^(m + 1)*((

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

(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQua

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

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

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

b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p

+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&

!ILtQ[m + 2*p + 1, 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] :> 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e} \\ & = \frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {10 \int \frac {\left (\frac {1}{2} \left (16 b c d-7 b^2 e-4 a c e\right )+8 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^3} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{21 c e^5} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{21 e^6}+\frac {\left (4 \left (-\frac {1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac {1}{4} c d (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{21 c e^6} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{21 c e^6 \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 (8 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac {1}{4} c d (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\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 )}{21 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{21 c e^6 \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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2+20 a c e^2\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 )}{21 c e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (veriﬁed)

Time = 34.04 (sec) , antiderivative size = 1375, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {d+e x} (a+x (b+c x))^{5/2} \left (\frac {2 \left (37 c^2 d^2-43 b c d e+9 b^2 e^2+16 a c e^2\right )}{21 e^5}+\frac {2 c (-4 c d+3 b e) x}{7 e^4}+\frac {2 c^2 x^2}{7 e^3}-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^2}-\frac {14 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)}\right )}{\left (a+b x+c x^2\right )^2}-\frac {(d+e x)^{3/2} (a+x (b+c x))^{5/2} \left (-4 (-2 c d+b e) \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (128 c^2 d^2+3 b^2 e^2+4 c e (-32 b d+29 a e)\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )+\frac {i \sqrt {2} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (128 c^2 d^2+3 b^2 e^2+4 c e (-32 b d+29 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}-\frac {i \sqrt {2} \left (-3 b^4 e^4+b^3 e^3 \left (32 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-2 b^2 c e^2 \left (16 c d^2+4 a e^2+67 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+8 c^2 \left (10 a^2 e^4-32 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (16 c d-29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+4 b c e \left (96 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-32 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{42 c e^7 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (a+b x+c x^2\right )^{5/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]

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

(Sqrt[d + e*x]*(a + x*(b + c*x))^(5/2)*((2*(37*c^2*d^2 - 43*b*c*d*e + 9*b^2*e^2 + 16*a*c*e^2))/(21*e^5) + (2*c

*(-4*c*d + 3*b*e)*x)/(7*e^4) + (2*c^2*x^2)/(7*e^3) - (2*(c*d^2 - b*d*e + a*e^2)^2)/(3*e^5*(d + e*x)^2) - (14*(

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

c*x))^(5/2)*(-4*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(128*

c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-32*b*d + 29*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d +

e*x)))/(d + e*x)) + (I*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 3*b^2*e^

2 + 4*c*e*(-32*b*d + 29*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) +

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

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

EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d +

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

I*Sqrt[2]*(-3*b^4*e^4 + b^3*e^3*(32*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2]) - 2*b^2*c*e^2*(16*c*d^2 + 4*a*e^2 + 67*d*

Sqrt[(b^2 - 4*a*c)*e^2]) + 8*c^2*(10*a^2*e^4 - 32*c*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*(16*c*d - 29*Sqrt[(b

^2 - 4*a*c)*e^2])) + 4*b*c*e*(96*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-32*c*d + 29*Sqrt[(b^2 - 4*a*c)*e^2]))

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

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

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

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

t[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(42*c*e^7*Sqrt[(c*d^2 + e*(-(

b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + b*x + c*x^2)^(5/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d

+ e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1869\) vs. \(2(552)=1104\).

Time = 3.23 (sec) , antiderivative size = 1870, normalized size of antiderivative = 3.01


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(1870\)
risch	\(\text {Expression too large to display}\)	\(2960\)
default	\(\text {Expression too large to display}\)	\(12847\)

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

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

^2*e^2-2*b*c*d^3*e+c^2*d^4)/e^7*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2-14/3*(c*e*x^2+b*e*x+

a*e)*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^6/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2/7*c^2/e^3

*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e

*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(3*c/e^4*(a*c*e^2+b^2*e^2-2*b*c*d*e+c^2*d^2)-2/7*c^2/e^

3*(5/2*a*e+5/2*b*d)-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*

x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*((3*a^2*c*e^4+3*a*b^2*e^4-12*a*b*c*d*e^3+9*a*c^2*d^2*e^2-2*b^3*d*e^3+9*b^

2*c*d^2*e^2-12*b*c^2*d^3*e+5*c^3*d^4)/e^6-1/3*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c*d^3*e+c^2*d

^4)/e^6*c-7/3*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^6*(b*e-c*d)+7/3*b/e^5*(a*b*e^3-2*a*c*d*e

^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*a*d-2/3*(3*c/e^4

*(a*c*e^2+b^2*e^2-2*b*c*d*e+c^2*d^2)-2/7*c^2/e^3*(5/2*a*e+5/2*b*d)-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b

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

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

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

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

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

2*d^2*e-4*c^3*d^3)+7/3*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^5*c-4/7*c^2/e^3*a*d-2/5*(c^2/e^

3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*(3/2*a*e+3/2*b*d)-2/3*(3*c/e^4*(a*c*e^2+b^2*e^2-2*b*c*d*e+c^2*d

^2)-2/7*c^2/e^3*(5/2*a*e+5/2*b*d)-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e

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

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

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

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

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

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

Fricas [C] (veriﬁcation not implemented)

Time = 0.14 (sec) , antiderivative size = 1091, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

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

2/63*((256*c^4*d^6 - 512*b*c^3*d^5*e + 2*(139*b^2*c^2 + 212*a*c^3)*d^4*e^2 - 2*(11*b^3*c + 212*a*b*c^2)*d^3*e^

3 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*d^2*e^4 + (256*c^4*d^4*e^2 - 512*b*c^3*d^3*e^3 + 2*(139*b^2*c^2 + 212*a

*c^3)*d^2*e^4 - 2*(11*b^3*c + 212*a*b*c^2)*d*e^5 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*e^6)*x^2 + 2*(256*c^4*d^

5*e - 512*b*c^3*d^4*e^2 + 2*(139*b^2*c^2 + 212*a*c^3)*d^3*e^3 - 2*(11*b^3*c + 212*a*b*c^2)*d^2*e^4 - (3*b^4 -

46*a*b^2*c - 120*a^2*c^2)*d*e^5)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/

(c^2*e^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)/(c^3*e^3), 1/

3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(256*c^4*d^5*e - 384*b*c^3*d^4*e^2 + 2*(67*b^2*c^2 + 116*a*c^3)*d^3*e^3 - (

3*b^3*c + 116*a*b*c^2)*d^2*e^4 + (256*c^4*d^3*e^3 - 384*b*c^3*d^2*e^4 + 2*(67*b^2*c^2 + 116*a*c^3)*d*e^5 - (3*

b^3*c + 116*a*b*c^2)*e^6)*x^2 + 2*(256*c^4*d^4*e^2 - 384*b*c^3*d^3*e^3 + 2*(67*b^2*c^2 + 116*a*c^3)*d^2*e^4 -

(3*b^3*c + 116*a*b*c^2)*d*e^5)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e

^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)/(c^3*e^3), weierstr

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

28*c^4*d^4*e^2 - 176*b*c^3*d^3*e^3 - 35*a*b*c^2*d*e^5 - 7*a^2*c^2*e^6 + (51*b^2*c^2 + 100*a*c^3)*d^2*e^4 - 3*(

2*c^4*d*e^5 - 3*b*c^3*e^6)*x^3 + (16*c^4*d^2*e^4 - 25*b*c^3*d*e^5 + (9*b^2*c^2 + 16*a*c^3)*e^6)*x^2 + (160*c^4

*d^3*e^3 - 224*b*c^3*d^2*e^4 - 49*a*b*c^2*e^6 + (67*b^2*c^2 + 130*a*c^3)*d*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(

e*x + d))/(c^2*e^9*x^2 + 2*c^2*d*e^8*x + c^2*d^2*e^7)

Sympy [F]

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

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

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

Maxima [F]

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

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

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

Giac [F]

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

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

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

Mupad [F(-1)]

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

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

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

\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [2457]

Optimal result