3.1263 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=516 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{15 d e^4 \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

[Out]

(-2*(d*(3*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 76*b*c*d*e + 15*b^2*e^2)) - c*e*(B*d*(16*c*d - 13*b*e) - 3*A
*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^4*(c*d - b*e)*Sqrt[d + e*x]) - (2*(d^2*(8*B*c*d - 5*b*B*e - 3*
A*c*e) + e*(B*d*(11*c*d - 8*b*e) - 3*A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(15*d*e^2*(c*d - b*e)*(d + e*x
)^(5/2)) + (2*Sqrt[-b]*Sqrt[c]*(3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(128*c^2*d^2 - 168*b*c*d*e + 4
3*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)]
)/(15*d*e^5*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(24*A*c*e*(2*c*d - b*e) - B*(128*c^
2*d^2 - 104*b*c*d*e + 15*b^2*e^2))*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*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.666191, antiderivative size = 516, 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 = {810, 812, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{15 d e^4 \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d*(3*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 76*b*c*d*e + 15*b^2*e^2)) - c*e*(B*d*(16*c*d - 13*b*e) - 3*A
*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^4*(c*d - b*e)*Sqrt[d + e*x]) - (2*(d^2*(8*B*c*d - 5*b*B*e - 3*
A*c*e) + e*(B*d*(11*c*d - 8*b*e) - 3*A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(15*d*e^2*(c*d - b*e)*(d + e*x
)^(5/2)) + (2*Sqrt[-b]*Sqrt[c]*(3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(128*c^2*d^2 - 168*b*c*d*e + 4
3*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)]
)/(15*d*e^5*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(24*A*c*e*(2*c*d - b*e) - B*(128*c^
2*d^2 - 104*b*c*d*e + 15*b^2*e^2))*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*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 810

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*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{2 \int \frac{\left (-\frac{1}{2} b d (8 B c d-5 b B e-3 A c e)-\frac{1}{2} c (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 d e^2 (c d-b e)}\\ &=-\frac{2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{15 d e^4 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{4 \int \frac{\frac{1}{4} b d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )+\frac{1}{4} c \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 d e^4 (c d-b e)}\\ &=-\frac{2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{15 d e^4 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{\left (24 A c e (2 c d-b e)-B \left (128 c^2 d^2-104 b c d e+15 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 e^5}+\frac{\left (c \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 d e^5 (c d-b e)}\\ &=-\frac{2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{15 d e^4 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{\left (\left (24 A c e (2 c d-b e)-B \left (128 c^2 d^2-104 b c d e+15 b^2 e^2\right )\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{15 e^5 \sqrt{b x+c x^2}}+\frac{\left (c \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{15 d e^5 (c d-b e) \sqrt{b x+c x^2}}\\ &=-\frac{2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{15 d e^4 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{\left (c \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\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 d e^5 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (\left (24 A c e (2 c d-b e)-B \left (128 c^2 d^2-104 b c d e+15 b^2 e^2\right )\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 e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{15 d e^4 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{2 \sqrt{-b} \sqrt{c} \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\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 d e^5 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} \left (24 A c e (2 c d-b e)-B \left (128 c^2 d^2-104 b c d e+15 b^2 e^2\right )\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 \sqrt{c} e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.10495, size = 530, normalized size = 1.03 \[ \frac{2 (x (b+c x))^{3/2} \left (e x \sqrt{\frac{b}{c}} (b+c x) \left ((d+e x)^2 \left (B d \left (23 b^2 e^2-93 b c d e+73 c^2 d^2\right )-3 A e \left (b^2 e^2-11 b c d e+11 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2-d (d+e x) (c d-b e) (6 A e (b e-2 c d)+B d (17 c d-11 b e))+5 B c d (d+e x)^3 (c d-b e)\right )+(d+e x)^2 \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) (3 A e (b e-8 c d)+4 B d (16 c d-7 b e)) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+B d \left (-43 b^2 e^2+168 b c d e-128 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+B d \left (-43 b^2 e^2+168 b c d e-128 c^2 d^2\right )\right )\right )\right )}{15 d e^5 x^2 \sqrt{\frac{b}{c}} (b+c x)^2 (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(Sqrt[b/c]*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 - d*(c*d - b*e)*(B*d*(17*c*d
- 11*b*e) + 6*A*e*(-2*c*d + b*e))*(d + e*x) + (-3*A*e*(11*c^2*d^2 - 11*b*c*d*e + b^2*e^2) + B*d*(73*c^2*d^2 -
93*b*c*d*e + 23*b^2*e^2))*(d + e*x)^2 + 5*B*c*d*(c*d - b*e)*(d + e*x)^3) + (d + e*x)^2*(Sqrt[b/c]*(B*d*(-128*c
^2*d^2 + 168*b*c*d*e - 43*b^2*e^2) + 3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2))*(b + c*x)*(d + e*x) + I*b*e*(B
*d*(-128*c^2*d^2 + 168*b*c*d*e - 43*b^2*e^2) + 3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2))*Sqrt[1 + b/(c*x)]*Sq
rt[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)*(4*B*d*(16*c*
d - 7*b*e) + 3*A*e*(-8*c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/S
qrt[x]], (c*d)/(b*e)])))/(15*Sqrt[b/c]*d*e^5*(c*d - b*e)*x^2*(b + c*x)^2*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.043, size = 4120, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(x*(c*x+b))^(1/2)*(-33*A*x^4*b*c^3*d*e^5+3*A*x^4*b^2*c^2*e^6-15*B*x*b^3*c*d^3*e^3-35*B*x^2*b^3*c*d^2*e^4+
24*A*x*b*c^3*d^4*e^2+76*B*x*b^2*c^2*d^4*e^2-64*B*x*b*c^3*d^5*e-18*B*x^4*b^2*c^2*d*e^5+103*B*x^4*b*c^3*d^2*e^4-
33*A*x^3*b^2*c^2*d*e^5-15*A*x^3*b*c^3*d^2*e^4-23*B*x^3*b^3*c*d*e^5+73*B*x^3*b^2*c^2*d^2*e^4+85*B*x^3*b*c^3*d^3
*e^3-48*A*x^2*b^2*c^2*d^2*e^4+33*A*x^2*b*c^3*d^3*e^3+158*B*x^2*b^2*c^2*d^3*e^3-68*B*x^2*b*c^3*d^4*e^2-21*A*x*b
^2*c^2*d^3*e^3+5*B*x^5*b*c^3*d*e^5+33*A*x^4*c^4*d^2*e^4-88*B*x^4*c^4*d^3*e^3+3*A*x^3*b^3*c*e^6+54*A*x^3*c^4*d^
3*e^3-144*B*x^3*c^4*d^4*e^2+24*A*x^2*c^4*d^4*e^2-64*B*x^2*c^4*d^5*e-5*B*x^5*c^4*d^2*e^4+128*B*EllipticE(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)-119*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/
(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+232*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*(
(c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-128*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-102*A*EllipticE(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)+192*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)-96*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e^2*((c*x+b)
/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-144*A*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+96*A*
EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)+422*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-592*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2
*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+256*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-238*B*EllipticF(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)+464*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-256*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-51*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+96*A*EllipticE(((c*x+b)/b)^
(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)
-48*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-
c*d))^(1/2)*(-c*x/b)^(1/2)+24*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-72*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2
*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*A*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+211*B
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)-296*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4
*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-43*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*A*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)-86*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)-51*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+96*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*
d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-48*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+24*A*EllipticF(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)-72*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c
*d))^(1/2)*(-c*x/b)^(1/2)+48*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+211*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d
^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-296*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-119*B*EllipticF(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)+232*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)+3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*e^6*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^2*
e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-43*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^4*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+128*B*EllipticE(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-128*B
*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)+15*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+30*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2))/(c*x+b)/x/(b*e-c*d)/(e*x+d)^(5/2)/c/d/e^5

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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{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B c x^{3} + A b x +{\left (B b + A c\right )} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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