3.426 \(\int \frac{1}{\sqrt{d+e x} (b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=451 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)} \]

[Out]

(-2*Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d +
e*x]*(b*(c*d - b*e)*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + 2*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)
*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (4*Sqrt[c]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*
Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7
/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sq
rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^
(7/2)*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.432825, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {740, 822, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d +
e*x]*(b*(c*d - b*e)*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + 2*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)
*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (4*Sqrt[c]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*
Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7
/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sq
rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^
(7/2)*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (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{1}{\sqrt{d+e x} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+\frac{3}{2} c e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{4 \int \frac{-\frac{1}{4} b c d e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-\frac{1}{2} c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}-\frac{\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{3 b^4 d (c d-b e) \sqrt{b x+c x^2}}-\frac{\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\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}{3 b^4 d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\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}{3 b^4 d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{4 \sqrt{c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\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 )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\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 )}{3 (-b)^{7/2} d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.43565, size = 429, normalized size = 0.95 \[ \frac{2 \left (b (d+e x) \left (b c^3 d^2 x^2 (c d-b e)+2 c^3 d^2 x^2 (b+c x) (4 c d-5 b e)-b d (b+c x)^2 (c d-b e)^2+2 x (b+c x)^2 (c d-b e)^2 (b e+4 c d)\right )-c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (3 b^2 c d e^2+2 b^3 e^3-13 b c^2 d^2 e+8 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 c d e^2+b^3 e^3-12 b c^2 d^2 e+8 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 b^2 c d e^2+b^3 e^3-12 b c^2 d^2 e+8 c^3 d^3\right )\right )\right )}{3 b^5 d^2 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]

[Out]

(2*(b*(d + e*x)*(b*c^3*d^2*(c*d - b*e)*x^2 + 2*c^3*d^2*(4*c*d - 5*b*e)*x^2*(b + c*x) - b*d*(c*d - b*e)^2*(b +
c*x)^2 + 2*(c*d - b*e)^2*(4*c*d + b*e)*x*(b + c*x)^2) - Sqrt[b/c]*c*x*(b + c*x)*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b
*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*
e^2 + b^3*e^3)*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*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*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)])))/(3*b^5*d^2*(c*d - b*e)^2*(x*(b + c*x))^(3/2)*Sqrt[d +
e*x])

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Maple [B]  time = 0.323, size = 1763, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(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))*x^2*b^4*c^2*d*e^3+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))*x^2*b^5*c*e^4+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))*x*b^6*e^4-b^3*c^3*d^4+9*x^3*b^3*c^3*d*e^3-33*x^3*b^2*c^4*d
^2*e^2+6*x^2*b^4*c^2*d*e^3-3*x^2*b^3*c^3*d^2*e^2-31*x^2*b^2*c^4*d^3*e-11*x*b^3*c^3*d^3*e-24*x^4*b*c^5*d^2*e^2-
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))*x^2*b^3*c^3*d^2*e^2+40*((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))*x^2*b^2*c^4*d^3*e+((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))*x^2*b^4*c^2*d*e^3+15*((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))*x^2*b^3*c^3*d^2*e^2-32*((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))*x^2*b^2
*c^4*d^3*e+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))*x*b^5*c*d*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))*x*b^4*c^2*d^2*e^2+40*((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))*x*b^3*c^3*d^3*e+((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))*x*b^5*c*d*e^3+15*((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))*x*b^4*c^2*d
^2*e^2-32*((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))*x*b^3*c^3*d^3*e+4*x*b^4*c^2*d^2*e^2+x*b^5*c*d*e^3+4*x^4*b^2*c^4*d*e^3+2*x^4*b^3*c^3*e^4+24*x^2*b*
c^5*d^4+4*x^3*b^4*c^2*e^4+6*x*b^2*c^4*d^4+16*x^4*c^6*d^3*e+16*x^3*c^6*d^4+2*x^2*b^5*c*e^4-16*((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))*x^2*b*c^5*d^4+
16*((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))*x^2*b*c^5*d^4-16*((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))*x*b^2*c^4*d^4+16*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^4*d^4-b^5*c*d^2*e^2+2*b^4*c^2*d^3*e)/x^2*(x*(c*x+b))^(1/
2)/d^2/b^4/c/(c*x+b)^2/(b*e-c*d)^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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(1/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((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{\sqrt{c x^{2} + b x} \sqrt{e x + d}}{c^{3} e x^{7} + b^{3} d x^{3} +{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{6} + 3 \,{\left (b c^{2} d + b^{2} c e\right )} x^{5} +{\left (3 \, b^{2} c d + b^{3} e\right )} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^3*e*x^7 + b^3*d*x^3 + (c^3*d + 3*b*c^2*e)*x^6 + 3*(b*c^2*d + b^2*c
*e)*x^5 + (3*b^2*c*d + b^3*e)*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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(1/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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