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

Optimal. Leaf size=666 \[ \frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-18 b^3 e^3-31 b c^2 d^2 e+16 c^3 d^3\right )}{9009 c^2 e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4-64 b c^3 d^3 e+32 c^4 d^4\right )+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-368 b c^4 d^4 e+128 c^5 d^5\right )}{9009 c^3 e^5}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{10 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x} (2 c d-b e)}{143 c e} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^5*d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c^2*d^2*e^3 - 17*b^4*c*d*e^4 +
24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 21*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*
x + c*x^2])/(9009*c^3*e^5) + (10*Sqrt[d + e*x]*(16*c^3*d^3 - 31*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3*e^3 - 14*
c*e*(c^2*d^2 - b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(9009*c^2*e^3) - (10*(2*c*d - b*e)*Sqrt[d + e*x]*(
b*x + c*x^2)^(5/2))/(143*c*e) + (2*(d + e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (4*Sqrt[-b]*(128*c^6*d^6 - 38
4*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*S
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/
2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3
*d^3*e + 79*b^2*c^2*d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellipti
cF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.832554, antiderivative size = 666, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {734, 832, 814, 843, 715, 112, 110, 117, 116} \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-18 b^3 e^3-31 b c^2 d^2 e+16 c^3 d^3\right )}{9009 c^2 e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4-64 b c^3 d^3 e+32 c^4 d^4\right )+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-368 b c^4 d^4 e+128 c^5 d^5\right )}{9009 c^3 e^5}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{10 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x} (2 c d-b e)}{143 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(128*c^5*d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c^2*d^2*e^3 - 17*b^4*c*d*e^4 +
24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 21*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*
x + c*x^2])/(9009*c^3*e^5) + (10*Sqrt[d + e*x]*(16*c^3*d^3 - 31*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3*e^3 - 14*
c*e*(c^2*d^2 - b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(9009*c^2*e^3) - (10*(2*c*d - b*e)*Sqrt[d + e*x]*(
b*x + c*x^2)^(5/2))/(143*c*e) + (2*(d + e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (4*Sqrt[-b]*(128*c^6*d^6 - 38
4*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*S
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/
2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3
*d^3*e + 79*b^2*c^2*d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellipti
cF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/2)*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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

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 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac{5 \int \sqrt{d+e x} (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2} \, dx}{13 e}\\ &=-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac{10 \int \frac{\left (\frac{1}{2} b d (c d+5 b e)+\left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{143 c e}\\ &=\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac{20 \int \frac{\left (-\frac{1}{4} b d \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3\right )-\frac{1}{4} \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{3003 c^2 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{9009 c^3 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac{8 \int \frac{\frac{1}{8} b d \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5\right )+\frac{1}{4} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{9009 c^3 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{9009 c^3 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{9009 c^3 e^6}-\frac{\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{9009 c^3 e^6}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{9009 c^3 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 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}{9009 c^3 e^6 \sqrt{b x+c x^2}}-\frac{\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{9009 c^3 e^6 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{9009 c^3 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac{\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\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}{9009 c^3 e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 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}{9009 c^3 e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{9009 c^3 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac{10 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac{4 \sqrt{-b} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\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 )}{9009 c^{7/2} e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{-b} d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 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 )}{9009 c^{7/2} e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.28284, size = 663, normalized size = 1. \[ \frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (b^2 c^3 e^2 \left (-218 d^2 e x+303 d^3+178 d e^2 x^2+1113 e^3 x^3\right )+b^3 c^2 e^3 \left (-22 d^2+12 d e x+15 e^2 x^2\right )-b^4 c e^4 (17 d+18 e x)+24 b^5 e^5+b c^4 e \left (-225 d^2 e^2 x^2+272 d^3 e x-368 d^4+196 d e^3 x^3+1701 e^4 x^4\right )+c^5 \left (80 d^3 e^2 x^2-70 d^2 e^3 x^3-96 d^4 e x+128 d^5+63 d e^4 x^4+693 e^5 x^5\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (383 b^2 c^4 d^4 e^2-70 b^3 c^3 d^3 e^3-25 b^4 c^2 d^2 e^4-64 b^5 c d e^5+48 b^6 e^6-400 b c^5 d^5 e+128 c^6 d^6\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6-384 b c^5 d^5 e+128 c^6 d^6\right )\right )\right )}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(24*b^5*e^5 - b^4*c*e^4*(17*d + 18*e*x) + b^3*c^2*e^3*(-22*d
^2 + 12*d*e*x + 15*e^2*x^2) + b^2*c^3*e^2*(303*d^3 - 218*d^2*e*x + 178*d*e^2*x^2 + 1113*e^3*x^3) + b*c^4*e*(-3
68*d^4 + 272*d^3*e*x - 225*d^2*e^2*x^2 + 196*d*e^3*x^3 + 1701*e^4*x^4) + c^5*(128*d^5 - 96*d^4*e*x + 80*d^3*e^
2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(128*c^6*d^6 - 384*b*c^5*d^5*e
+ 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*(b + c*x)*(d +
e*x) - (2*I)*b*e*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e
^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)] + I*b*e*(128*c^6*d^6 - 400*b*c^5*d^5*e + 383*b^2*c^4*d^4*e^2 - 70*b^3*c^3*d^3*e^3 - 25*b^
4*c^2*d^2*e^4 - 64*b^5*c*d*e^5 + 48*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(9009*b*c^3*e^6*x^3*(b + c*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.287, size = 1728, normalized size = 2.6 \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]

2/9009*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-23*((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^6*c^2*d^2*e^5+303*x*b^3*c^5*d^4*e^3-368*x*b^2*c^6*d^5*e^2+128
*x*b*c^7*d^6*e+2653*x^6*b*c^7*d*e^6+48*((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^8*e^7+24*x*b^6*c^2*d*e^6-17*x*b^5*c^3*d^2*e^5+207*x^2*b^2*c^6*d^4*
e^3+3188*x^5*b^2*c^6*d*e^6-36*x^5*b*c^7*d^2*e^5+1318*x^4*b^3*c^5*d*e^6-69*x^4*b^2*c^6*d^2*e^5+57*x^4*b*c^7*d^3
*e^4-8*x^3*b^4*c^4*d*e^6-50*x^3*b^3*c^5*d^2*e^5+132*x^3*b^2*c^6*d^3*e^4-112*x^3*b*c^7*d^4*e^3-11*x^2*b^5*c^3*d
*e^6-27*x^2*b^4*c^4*d^2*e^5+63*x^2*b^3*c^5*d^3*e^4-336*x^2*b*c^7*d^5*e^2+778*((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^4*d^4*e^3-1454*((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^
5*d^5*e^2+1024*((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^6*d^6*e+24*((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^7*c*d*e^6-20*((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^5*c^3*d^3*e^4-395*((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^4*d^4*e^3+1054*((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^5
*d^5*e^2-896*((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^6*d^6*e-88*((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*c*d*e^6-2*((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^2*d^2*e^5-50*((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^3*d^3*e^4+693*x^8*c^8*e^7-22*
x*b^4*c^4*d^3*e^4+2394*x^7*b*c^7*e^7+756*x^7*c^8*d*e^6+2814*x^6*b^2*c^6*e^7-7*x^6*c^8*d^2*e^5+1128*x^5*b^3*c^5
*e^7+10*x^5*c^8*d^3*e^4-3*x^4*b^4*c^4*e^7-16*x^4*c^8*d^4*e^3+6*x^3*b^5*c^3*e^7+32*x^3*c^8*d^5*e^2+24*x^2*b^6*c
^2*e^7+128*x^2*c^8*d^6*e+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^7*d^7-256*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^7*d^7)/c^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)/e^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 ({\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{\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)