\(\int \frac {(b x+c x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [400]

Optimal result

Integrand size = 23, antiderivative size = 537 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

10/693*(16*c^2*d^2-23*b*c*d*e+3*b^2*e^2-7*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/c/e^3+2/11*(c*x^
2+b*x)^(5/2)*(e*x+d)^(1/2)/e-2/693*(-b*e+2*c*d)*(8*b^4*e^4+29*b^3*c*d*e^3+99*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+1
28*c^4*d^4)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(
1/2)/c^(5/2)/e^6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+4/693*d*(-b*e+c*d)*(2*b^4*e^4+5*b^3*c*d*e^3+123*b^2*c^2*d^2
*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+
c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(5/2)/e^6/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/693*(128*c^4*d^4-304*b*c^3*d^3*e+19
5*b^2*c^2*d^2*e^2-7*b^3*c*d*e^3-4*b^4*e^4-12*c*e*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*x)*(e*x+d)^(1/2)*
(c*x^2+b*x)^(1/2)/c^2/e^5

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {748, 828, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^4 e^4+29 b^3 c d e^3+99 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^4 e^4-7 b^3 c d e^3-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e} \]

[In]

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

[Out]

(2*Sqrt[d + e*x]*(128*c^4*d^4 - 304*b*c^3*d^3*e + 195*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - 4*b^4*e^4 - 12*c*e*(2*
c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(693*c^2*e^5) + (10*Sqrt[d + e*x]*(16*c^2*d
^2 - 23*b*c*d*e + 3*b^2*e^2 - 7*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(693*c*e^3) + (2*Sqrt[d + e*x]*(b*x
+ c*x^2)^(5/2))/(11*e) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 99*b^2*c^2*d^2*e^2 + 29*b^
3*c*d*e^3 + 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (
b*e)/(c*d)])/(693*c^(5/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d - b*e)*(128*c^4*d^4 -
256*b*c^3*d^3*e + 123*b^2*c^2*d^2*e^2 + 5*b^3*c*d*e^3 + 2*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]
*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(693*c^(5/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x]*(Sqrt[
1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)])), Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 729

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b
*x + c*x^2]), Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 748

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 + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 828

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])

Rule 857

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]
&&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \int \frac {(b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 e} \\ & = \frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {10 \int \frac {\left (-\frac {1}{2} b d \left (16 c^2 d^2-23 b c d e+3 b^2 e^2\right )-2 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{231 c e^3} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {4 \int \frac {\frac {1}{4} b d \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4\right )+\frac {1}{4} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{693 c^2 e^5} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{693 c^2 e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{693 c^2 e^6} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{693 c^2 e^6 \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{693 c^2 e^6 \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{693 c^2 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{693 c^2 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.34 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.04 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (-4 b^4 e^4+b^3 c e^3 (-7 d+3 e x)+b^2 c^2 e^2 \left (195 d^2-139 d e x+113 e^2 x^2\right )+b c^3 e \left (-304 d^3+224 d^2 e x-185 d e^2 x^2+161 e^3 x^3\right )+c^4 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (-256 c^5 d^5+640 b c^4 d^4 e-454 b^2 c^3 d^3 e^2+41 b^3 c^2 d^2 e^3+13 b^4 c d e^4+8 b^5 e^5\right ) (b+c x) (d+e x)-i b e \left (256 c^5 d^5-640 b c^4 d^4 e+454 b^2 c^3 d^3 e^2-41 b^3 c^2 d^2 e^3-13 b^4 c d e^4-8 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (128 c^5 d^5-336 b c^4 d^4 e+259 b^2 c^3 d^3 e^2-34 b^3 c^2 d^2 e^3-9 b^4 c d e^4-8 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^4*e^4 + b^3*c*e^3*(-7*d + 3*e*x) + b^2*c^2*e^2*(195*d^
2 - 139*d*e*x + 113*e^2*x^2) + b*c^3*e*(-304*d^3 + 224*d^2*e*x - 185*d*e^2*x^2 + 161*e^3*x^3) + c^4*(128*d^4 -
96*d^3*e*x + 80*d^2*e^2*x^2 - 70*d*e^3*x^3 + 63*e^4*x^4)) + Sqrt[b/c]*(Sqrt[b/c]*(-256*c^5*d^5 + 640*b*c^4*d^
4*e - 454*b^2*c^3*d^3*e^2 + 41*b^3*c^2*d^2*e^3 + 13*b^4*c*d*e^4 + 8*b^5*e^5)*(b + c*x)*(d + e*x) - I*b*e*(256*
c^5*d^5 - 640*b*c^4*d^4*e + 454*b^2*c^3*d^3*e^2 - 41*b^3*c^2*d^2*e^3 - 13*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/
(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(128*c^5*d^5 - 3
36*b*c^4*d^4*e + 259*b^2*c^3*d^3*e^2 - 34*b^3*c^2*d^2*e^3 - 9*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[
1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(693*b*c^2*e^6*x^3*(b + c*x)^3*Sq
rt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1440\) vs. \(2(477)=954\).

Time = 1.95 (sec) , antiderivative size = 1441, normalized size of antiderivative = 2.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1441\)
elliptic	\(\text {Expression too large to display}\)	\(1442\)

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/693*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(31*b*c^6*d*e^5*x^5+50*b^2*c^5*d*e^5*x^4-49*b*c^6*d^2*e^4*x^4+30*b^3*c^
4*d*e^5*x^3-95*b^2*c^5*d^2*e^4*x^3+96*b*c^6*d^3*e^3*x^3+8*b^4*c^3*d*e^5*x^2-49*b^3*c^4*d^2*e^4*x^2-115*b^2*c^5
*d^3*e^3*x^2+272*b*c^6*d^4*e^2*x^2+4*b^5*c^2*d*e^5*x+7*b^4*c^3*d^2*e^4*x-195*b^3*c^4*d^3*e^3*x+304*b^2*c^5*d^4
*e^2*x-128*b*c^6*d^5*e*x-224*b*c^6*e^6*x^6+7*c^7*d*e^5*x^6-274*b^2*c^5*e^6*x^5-10*c^7*d^2*e^4*x^5-116*b^3*c^4*
e^6*x^4+16*c^7*d^3*e^3*x^4+b^4*c^3*e^6*x^3-32*c^7*d^4*e^2*x^3+4*b^5*c^2*e^6*x^2-128*c^7*d^5*e*x^2-256*((c*x+b)
/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6
*d^6+256*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b*c^6*d^6-63*c^7*e^6*x^7+4*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c*d*e^5+6*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^4+236*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^3-758*((c*x+b)
/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c
^4*d^4*e^2+768*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*b^2*c^5*d^5*e+5*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c*d*e^5+28*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^4+1094*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^4*e^2-896*((c*x+b)/b)
^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^5*
d^5*e+8*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*b^7*e^6-495*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^3)/c^4/e^6/x/(c*e*x^2+b*e*x+c*d*x+b*d)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.27 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.19 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left ({\left (256 \, c^{6} d^{6} - 768 \, b c^{5} d^{5} e + 726 \, b^{2} c^{4} d^{4} e^{2} - 172 \, b^{3} c^{3} d^{3} e^{3} - 33 \, b^{4} c^{2} d^{2} e^{4} - 9 \, b^{5} c d e^{5} - 8 \, b^{6} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (256 \, c^{6} d^{5} e - 640 \, b c^{5} d^{4} e^{2} + 454 \, b^{2} c^{4} d^{3} e^{3} - 41 \, b^{3} c^{3} d^{2} e^{4} - 13 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (63 \, c^{6} e^{6} x^{4} + 128 \, c^{6} d^{4} e^{2} - 304 \, b c^{5} d^{3} e^{3} + 195 \, b^{2} c^{4} d^{2} e^{4} - 7 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6} - 7 \, {\left (10 \, c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} x^{3} + {\left (80 \, c^{6} d^{2} e^{4} - 185 \, b c^{5} d e^{5} + 113 \, b^{2} c^{4} e^{6}\right )} x^{2} - {\left (96 \, c^{6} d^{3} e^{3} - 224 \, b c^{5} d^{2} e^{4} + 139 \, b^{2} c^{4} d e^{5} - 3 \, b^{3} c^{3} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{2079 \, c^{4} e^{7}} \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/2079*((256*c^6*d^6 - 768*b*c^5*d^5*e + 726*b^2*c^4*d^4*e^2 - 172*b^3*c^3*d^3*e^3 - 33*b^4*c^2*d^2*e^4 - 9*b^
5*c*d*e^5 - 8*b^6*e^6)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3
*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(256*c^6*d^5
*e - 640*b*c^5*d^4*e^2 + 454*b^2*c^4*d^3*e^3 - 41*b^3*c^3*d^2*e^4 - 13*b^4*c^2*d*e^5 - 8*b^5*c*e^6)*sqrt(c*e)*
weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2
+ 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*
b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(63*c^6*e^6*x^4 + 12
8*c^6*d^4*e^2 - 304*b*c^5*d^3*e^3 + 195*b^2*c^4*d^2*e^4 - 7*b^3*c^3*d*e^5 - 4*b^4*c^2*e^6 - 7*(10*c^6*d*e^5 -
23*b*c^5*e^6)*x^3 + (80*c^6*d^2*e^4 - 185*b*c^5*d*e^5 + 113*b^2*c^4*e^6)*x^2 - (96*c^6*d^3*e^3 - 224*b*c^5*d^2
*e^4 + 139*b^2*c^4*d*e^5 - 3*b^3*c^3*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^4*e^7)

Sympy [F]

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

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(1/2),x)

[Out]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(1/2), x)