3.1261 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=449 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(7*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 60*b*c*d*e + b^2*e^2) + 3*c*e*(16*B*c*d - b*B*e -
14*A*c*e)*x)*Sqrt[b*x + c*x^2])/(35*c*e^4) + (2*(8*B*d - 7*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d +
e*x]) + (2*Sqrt[-b]*(5*b*c*e*(8*B*d - 7*A*e)*(2*c*d - b*e) - (16*B*c*d - b*B*e - 14*A*c*e)*(8*c^2*d^2 - 3*b*c*
d*e - 2*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)])/(35*c^(3/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(56*A*c*e*(2*c*d - b*
e) - B*(128*c^2*d^2 - 72*b*c*d*e - b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.56866, antiderivative size = 449, 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 = {812, 814, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(7*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 60*b*c*d*e + b^2*e^2) + 3*c*e*(16*B*c*d - b*B*e -
14*A*c*e)*x)*Sqrt[b*x + c*x^2])/(35*c*e^4) + (2*(8*B*d - 7*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d +
e*x]) + (2*Sqrt[-b]*(5*b*c*e*(8*B*d - 7*A*e)*(2*c*d - b*e) - (16*B*c*d - b*B*e - 14*A*c*e)*(8*c^2*d^2 - 3*b*c*
d*e - 2*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)])/(35*c^(3/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(56*A*c*e*(2*c*d - b*
e) - B*(128*c^2*d^2 - 72*b*c*d*e - b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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 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 \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}-\frac{6 \int \frac{\left (\frac{1}{2} b (8 B d-7 A e)+\frac{1}{2} (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 e^2}\\ &=-\frac{2 \sqrt{d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{35 c e^4}+\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}+\frac{4 \int \frac{\frac{1}{4} b d \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )\right )+\frac{1}{4} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-2 (16 B c d-b B e-14 A c e) \left (4 c^2 d^2-\frac{3}{2} b c d e-b^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{35 c e^4}\\ &=-\frac{2 \sqrt{d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{35 c e^4}+\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}+\frac{\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{35 c e^5}-\frac{\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{35 c e^5}\\ &=-\frac{2 \sqrt{d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{35 c e^4}+\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}+\frac{\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 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}{35 c e^5 \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-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}{35 c e^5 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{35 c e^4}+\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}+\frac{\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 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}{35 c e^5 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-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}{35 c e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt{b x+c x^2}}{35 c e^4}+\frac{2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt{d+e x}}+\frac{2 \sqrt{-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 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 )}{35 c^{3/2} e^5 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-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 )}{35 c^{3/2} e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.54842, size = 514, normalized size = 1.14 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left ((d+e x) \left (7 A c e (2 b e-3 c d)+B \left (b^2 e^2-25 b c d e+29 c^2 d^2\right )\right )+c e x (d+e x) (7 A c e+8 b B e-13 B c d)+35 c d (B d-A e) (c d-b e)+5 B c^2 e^2 x^2 (d+e x)\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (7 A c e (8 c d-b e)+2 B \left (b^2 e^2+6 b c d e-32 c^2 d^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (7 A c e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (11 b^2 c d e^2+2 b^3 e^3-136 b c^2 d^2 e+128 c^3 d^3\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (7 A c e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (11 b^2 c d e^2+2 b^3 e^3-136 b c^2 d^2 e+128 c^3 d^3\right )\right )\right )\right )}{35 b c e^5 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(35*c*d*(B*d - A*e)*(c*d - b*e) + (7*A*c*e*(-3*c*d + 2*b*e) + B*(29*c^
2*d^2 - 25*b*c*d*e + b^2*e^2))*(d + e*x) + c*e*(-13*B*c*d + 8*b*B*e + 7*A*c*e)*x*(d + e*x) + 5*B*c^2*e^2*x^2*(
d + e*x)) + Sqrt[b/c]*(Sqrt[b/c]*(7*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*(128*c^3*d^3 - 136*b*c^2*d^2
*e + 11*b^2*c*d*e^2 + 2*b^3*e^3))*(b + c*x)*(d + e*x) + I*b*e*(7*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B
*(128*c^3*d^3 - 136*b*c^2*d^2*e + 11*b^2*c*d*e^2 + 2*b^3*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ell
ipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(7*A*c*e*(8*c*d - b*e) + 2*B*(-32*c^2*d^
2 + 6*b*c*d*e + b^2*e^2))*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)])))/(35*b*c*e^5*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.035, size = 1610, normalized size = 3.6 \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)*(c*x^2+b*x)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-2/35*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-13*B*x^4*b*c^4*e^4+200*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*e-9*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^3-264*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*e+112*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*e-119*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e^3+224*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^2-112*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*e-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^3-71*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^2+56*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
F(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e^3-168*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^2+60*B*x*b^2*c^3*d^2*e^2-64*B*
x*b*c^4*d^3*e+44*B*x^2*b*c^4*d^2*e^2-49*A*x*b^2*c^3*d*e^3+56*A*x*b*c^4*d^2*e^2+25*B*x^3*b*c^4*d*e^3-35*A*x^2*b
*c^4*d*e^3+16*B*x^2*b^2*c^3*d*e^3-5*B*x^5*c^5*e^4-7*A*x^4*c^5*e^4-B*x*b^3*c^2*d*e^3+8*B*x^4*c^5*d*e^3-21*A*x^3
*b*c^4*e^4+14*A*x^3*c^5*d*e^3-9*B*x^3*b^2*c^3*e^4-16*B*x^3*c^5*d^2*e^2-14*A*x^2*b^2*c^3*e^4+56*A*x^2*c^5*d^2*e
^2-B*x^2*b^3*c^2*e^4-64*B*x^2*c^5*d^3*e-128*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^4+128*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*c^4*d^4+7*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^4+147*B*EllipticE
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)-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))*b^5*e^4)/c^3/e^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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