3.1272 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=399 \[ -\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right )}{3 b^2 c^2} \]

[Out]

(-2*(d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*Sqrt[b*x + c*x^2]) + (2*e*(6
*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*(6*A*c^3*d^2 - 8*b
^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Ellipti
cE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]
) - (2*d*(c*d - b*e)*(6*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^
2])

________________________________________________________________________________________

Rubi [A]  time = 0.547967, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {818, 832, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right )}{3 b^2 c^2}-\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*Sqrt[b*x + c*x^2]) + (2*e*(6
*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*(6*A*c^3*d^2 - 8*b
^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Ellipti
cE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]
) - (2*d*(c*d - b*e)*(6*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^
2])

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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 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{(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 \int \frac{\sqrt{d+e x} \left (\frac{1}{2} b (b B+3 A c) d e+\frac{1}{2} e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{4 \int \frac{\frac{1}{4} b d e \left (3 A c^2 d-4 b^2 B e+3 b c (2 B d+A e)\right )+\frac{1}{4} e \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 c^2}\\ &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}-\frac{\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 c^2}+\frac{\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^2 c^2}\\ &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}-\frac{\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{3 b^2 c^2 \sqrt{b x+c x^2}}+\frac{\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{3 b^2 c^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\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^2 c^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\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^2 c^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\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)^{3/2} c^{5/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\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)^{3/2} c^{5/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.11929, size = 391, normalized size = 0.98 \[ \frac{2 \left (b (d+e x) \left (3 x (b B-A c) (c d-b e)^2-3 A c^2 d^2 (b+c x)+b^2 B e^2 x (b+c x)\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (-3 b c (2 A e+3 B d)+3 A c^2 d+8 b^2 B e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right )\right )\right )}{3 b^3 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(2*(b*(d + e*x)*(3*(b*B - A*c)*(c*d - b*e)^2*x - 3*A*c^2*d^2*(b + c*x) + b^2*B*e^2*x*(b + c*x)) + Sqrt[b/c]*(S
qrt[b/c]*(6*A*c^3*d^2 - 8*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*(b + c*x)*(d + e*x)
+ I*b*e*(6*A*c^3*d^2 - 8*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*Sqrt[1 + b/(c*x)]*Sqr
t[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(3*A*c^2*d + 8
*b^2*B*e - 3*b*c*(3*B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)])))/(3*b^3*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 1324, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2/3*(-3*A*x*b*c^4*d^2*e-3*B*x*b*c^4*d^3-4*B*x*b^3*c^2*d*e^2-6*A*x^2*b*c^4*d*e^2-3*B*x^2*b*c^4*d^2*e+3*A*x*b^2
*c^3*d*e^2+6*B*x*b^2*c^3*d^2*e+5*B*x^2*b^2*c^3*d*e^2+3*A*b*c^4*d^3+6*A*x*c^5*d^3+3*A*x^2*b^2*c^3*e^3+6*A*x^2*c
^5*d^2*e-B*x^3*b^2*c^3*e^3-4*B*x^2*b^3*c^2*e^3-6*A*((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^4*d^3+6*A*((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^4*d^3+3*B*((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^3*d^3-3*B*((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^3*d^3+6*A*((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*e^3-8*B*((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*e^3-12*A*((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^3*c^2*d*e^2+12*A*((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^3*d^2*e+3*A*((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^2*d*e^2-9*A*
((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^3*d^2*e+21*B*((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*d*e^2-16*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e-4*B*((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*d*e^2+7*B*((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^2*d^2*e)/x*(x*(c*x+
b))^(1/2)/(c*x+b)/c^4/b^2/(e*x+d)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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((B*x+A)*(e*x+d)**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(3/2), x)