3.1271 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=527 \[ -\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c^2 d e (95 A e+103 B d)-8 b^3 c e^2 (5 A e+16 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2} \]

[Out]

(-2*(d + e*x)^(5/2)*(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*(3
0*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*
x^2])/(15*b^2*c^3) + (2*e*(10*A*c^2*d + 6*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*b
^2*c^2) + (2*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e) - 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^
2*d*e*(103*B*d + 95*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*d*(c*d - b*e)*(30*A*c^3*d^2 -
24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)
/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[d + e*x]*Sqrt[b*x
+ c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.906669, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}-\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c^2 d e (95 A e+103 B d)-8 b^3 c e^2 (5 A e+16 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(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*(3
0*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*
x^2])/(15*b^2*c^3) + (2*e*(10*A*c^2*d + 6*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*b
^2*c^2) + (2*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e) - 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^
2*d*e*(103*B*d + 95*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*d*(c*d - b*e)*(30*A*c^3*d^2 -
24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)
/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*(-b)^(3/2)*c^(7/2)*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)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{5/2} \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{(d+e x)^{3/2} \left (\frac{1}{2} b (b B+5 A c) d e+\frac{1}{2} e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac{2 (d+e x)^{5/2} \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 (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{1}{4} b d e \left (15 A c^2 d-6 b^2 B e+5 b c (2 B d+A e)\right )+\frac{1}{4} e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{5 b^2 c^2}\\ &=-\frac{2 (d+e x)^{5/2} \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 (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 b^2 c^3}+\frac{2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}+\frac{8 \int \frac{\frac{1}{8} b d e \left (15 A c^3 d^2+24 b^3 B e^2+45 b c^2 d (B d+A e)-b^2 c e (61 B d+20 A e)\right )+\frac{1}{8} e \left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 b^2 c^3}\\ &=-\frac{2 (d+e x)^{5/2} \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 (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 b^2 c^3}+\frac{2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}-\frac{\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 b^2 c^3}+\frac{\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 b^2 c^3}\\ &=-\frac{2 (d+e x)^{5/2} \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 (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 b^2 c^3}+\frac{2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}-\frac{\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{15 b^2 c^3 \sqrt{b x+c x^2}}+\frac{\left (\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{15 b^2 c^3 \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2} \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 (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 b^2 c^3}+\frac{2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}+\frac{\left (\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 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}{15 b^2 c^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 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}{15 b^2 c^3 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2} \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 (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 b^2 c^3}+\frac{2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 b^2 c^2}+\frac{2 \left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 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 )}{15 (-b)^{3/2} c^{7/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 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 )}{15 (-b)^{3/2} c^{7/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 4.11293, size = 493, normalized size = 0.94 \[ \frac{2 \left (b (d+e x) \left (b^2 e^2 x (b+c x) (5 A c e-9 b B e+16 B c d)+15 x (b B-A c) (c d-b e)^3-15 A c^3 d^3 (b+c x)+3 b^2 B c e^3 x^2 (b+c x)\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (8 b^2 c e (5 A e+13 B d)-15 b c^2 d (5 A e+4 B d)+15 A c^3 d^2-48 b^3 B e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 c^2 d e (95 A e+103 B d)-8 b^3 c e^2 (5 A e+16 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 c^2 d e (95 A e+103 B d)-8 b^3 c e^2 (5 A e+16 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right )\right )\right )}{15 b^3 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(2*(b*(d + e*x)*(15*(b*B - A*c)*(c*d - b*e)^3*x - 15*A*c^3*d^3*(b + c*x) + b^2*e^2*(16*B*c*d - 9*b*B*e + 5*A*c
*e)*x*(b + c*x) + 3*b^2*B*c*e^3*x^2*(b + c*x)) + Sqrt[b/c]*(Sqrt[b/c]*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*
d^2*(B*d + 3*A*e) - 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B*d + 95*A*e))*(b + c*x)*(d + e*x) + I*b*e
*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e) - 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B
*d + 95*A*e))*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*(c*d - b*e)*(15*A*c^3*d^2 - 48*b^3*B*e^2 - 15*b*c^2*d*(4*B*d + 5*A*e) + 8*b^2*c*e*(13*B*d + 5*A*e))*S
qrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b^3*c^3
*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.069, size = 1766, normalized size = 3.4 \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)^(7/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/15*(45*A*x^2*b*c^5*d^2*e^2+15*B*x*b*c^5*d^4+15*B*x^2*b*c^5*d^3*e-45*A*x*b^2*c^4*d^2*e^2-45*B*x*b^2*c^4*d^3*e
+30*A*x*b*c^5*d^3*e+55*B*x^2*b^3*c^3*d*e^3-24*B*x*b^4*c^2*d*e^3-40*A*x^2*b^2*c^4*d*e^3+20*A*x*b^3*c^3*d*e^3+19
*B*x^3*b^2*c^4*d*e^3-29*B*x^2*b^2*c^4*d^2*e^2+61*B*x*b^3*c^3*d^2*e^2-15*A*b*c^5*d^4-30*A*x*c^6*d^4+3*B*x^4*b^2
*c^4*e^4-30*A*x^2*c^6*d^3*e-6*B*x^3*b^3*c^3*e^4-24*B*x^2*b^4*c^2*e^4+5*A*x^3*b^2*c^4*e^4+20*A*x^2*b^3*c^3*e^4+
60*e*b^2*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))*c^4*d^3+176*e^3*b^5*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))*c*d-231*e^2*b^4*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))*c^2*d^2+118*e*b^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))*c^3*d^3-24*e^3*b^5*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))*c*d
+67*e^2*b^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))*c^2*d^2-58*e*b^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))*c^3*d^3-135*e^3*b^4*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))*c^2*d+140*e^2*b^3*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))*c^3*d^2+30*b*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))*c^5*d
^4-48*e^4*b^6*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))+40*e^4*b^5*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))*c-30*b*A*((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))*c^5*d^4-15*b^2*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))*c^4*d^4+15*b^2*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))*c^4*d^4-75*e*b^2*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))*c
^4*d^3+20*e^3*b^4*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))*c^2*d-50*e^2*b^3*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*c^3*d^2)/x*(x*(c*x+b))^(1/2)/(c*x+b)/c^5/b^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{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)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A d^{3} +{\left (3 \, B d e^{2} + A e^{3}\right )} x^{3} + 3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{2} +{\left (B d^{3} + 3 \, A d^{2} 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)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^4 + A*d^3 + (3*B*d*e^2 + A*e^3)*x^3 + 3*(B*d^2*e + A*d*e^2)*x^2 + (B*d^3 + 3*A*d^2*e)*x)*sqr
t(c*x^2 + b*x)*sqrt(e*x + d)/(c^2*x^4 + 2*b*c*x^3 + b^2*x^2), 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((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{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)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)