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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.603732, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {734, 814, 843, 715, 112, 110, 117, 116} \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

Antiderivative was successfully verified.

[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 734

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 814

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 843

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]

Rule 715

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 112

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 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, 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 117

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 116

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

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\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]  time = 2.87561, size = 557, normalized size = 1.04 \[ \frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (b^2 c^2 e^2 \left (195 d^2-139 d e x+113 e^2 x^2\right )+b^3 c e^3 (3 e x-7 d)-4 b^4 e^4+b c^3 e \left (224 d^2 e x-304 d^3-185 d e^2 x^2+161 e^3 x^3\right )+c^4 \left (80 d^2 e^2 x^2-96 d^3 e x+128 d^4-70 d e^3 x^3+63 e^4 x^4\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (259 b^2 c^3 d^3 e^2-34 b^3 c^2 d^2 e^3-9 b^4 c d e^4-8 b^5 e^5-336 b c^4 d^4 e+128 c^5 d^5\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (454 b^2 c^3 d^3 e^2-41 b^3 c^2 d^2 e^3-13 b^4 c d e^4-8 b^5 e^5-640 b c^4 d^4 e+256 c^5 d^5\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-454 b^2 c^3 d^3 e^2+41 b^3 c^2 d^2 e^3+13 b^4 c d e^4+8 b^5 e^5+640 b c^4 d^4 e-256 c^5 d^5\right )\right )\right )}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.279, size = 1441, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/693*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-224*x^6*b*c^6*e^6+7*x^6*c^7*d*e^5-274*x^5*b^2*c^5*e^6-10*x^5*c^7*d^2*
e^4-116*x^4*b^3*c^4*e^6+16*x^4*c^7*d^3*e^3+x^3*b^4*c^3*e^6-32*x^3*c^7*d^4*e^2+4*x^2*b^5*c^2*e^6-128*x^2*c^7*d^
5*e+272*x^2*b*c^6*d^4*e^2+4*x*b^5*c^2*d*e^5+7*x*b^4*c^3*d^2*e^4-195*x*b^3*c^4*d^3*e^3+304*x*b^2*c^5*d^4*e^2-12
8*x*b*c^6*d^5*e+31*x^5*b*c^6*d*e^5+50*x^4*b^2*c^5*d*e^5-49*x^4*b*c^6*d^2*e^4+30*x^3*b^3*c^4*d*e^5-95*x^3*b^2*c
^5*d^2*e^4+96*x^3*b*c^6*d^3*e^3+8*x^2*b^4*c^3*d*e^5-49*x^2*b^3*c^4*d^2*e^4+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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)*
(-(e*x+d)*c/(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
-256*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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+5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^5-63*x^7*c^7*e^6-115*x^2*b^2*c^5*d^3*e^3+256*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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+768*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(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-758*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticF(((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)*(-(e*x+d)*c/(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+28*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(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+10
94*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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)*(-(e*x+d)*c/(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)/c^4/e^6/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}{\sqrt{e x + d}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)/sqrt(e*x + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)