3.1259 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=494 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (A e (2 c d-b e)+B d (8 c d-9 b e)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 d e^3 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )}{15 d^2 e^2 \sqrt{d+e x} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} (e x (B d (7 c d-6 b e)-A e (2 c d-b e))+d (A e (c d-2 b e)+B d (4 c d-3 b e)))}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

[Out]

(2*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^2
*e^2*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*(d*(B*d*(4*c*d - 3*b*e) + A*e*(c*d - 2*b*e)) + e*(B*d*(7*c*d - 6*b*e) -
A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2)) - (2*Sqrt[-b]*Sqrt[c]*(2*A*e*
(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 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^2*e^3*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sq
rt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(B*d*(8*c*d - 9*b*e) + A*e*(2*c*d - b*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d*e^3*(c*d - b*e)*Sqrt[d + e*x]
*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.637924, antiderivative size = 494, 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, 834, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{b x+c x^2} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )}{15 d^2 e^2 \sqrt{d+e x} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} (e x (B d (7 c d-6 b e)-A e (2 c d-b e))+d (A e (c d-2 b e)+B d (4 c d-3 b e)))}{15 d e^2 (d+e x)^{5/2} (c d-b e)}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (A e (2 c d-b e)+B d (8 c d-9 b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^3 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^2
*e^2*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*(d*(B*d*(4*c*d - 3*b*e) + A*e*(c*d - 2*b*e)) + e*(B*d*(7*c*d - 6*b*e) -
A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2)) - (2*Sqrt[-b]*Sqrt[c]*(2*A*e*
(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 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^2*e^3*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sq
rt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(B*d*(8*c*d - 9*b*e) + A*e*(2*c*d - b*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d*e^3*(c*d - b*e)*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^{7/2}} \, dx &=-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{2 \int \frac{\frac{1}{2} b \left (2 b e \left (\frac{3 B d}{2}+A e\right )-c d (4 B d+A e)\right )-\frac{1}{2} c (2 c d (4 B d+A e)-b e (9 B d+A e)) x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{15 d e^2 (c d-b e)}\\ &=\frac{2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{4 \int \frac{-\frac{1}{4} b c d \left (A e (c d+b e)+B \left (4 c d^2-6 b d e\right )\right )-\frac{1}{4} c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 d^2 e^2 (c d-b e)^2}\\ &=\frac{2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{(c (B d (8 c d-9 b e)+A e (2 c d-b e))) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 d e^3 (c d-b e)}-\frac{\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 d^2 e^3 (c d-b e)^2}\\ &=\frac{2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac{\left (c (B d (8 c d-9 b e)+A e (2 c d-b e)) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{15 d e^3 (c d-b e) \sqrt{b x+c x^2}}-\frac{\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt{b x+c x^2}}\\ &=\frac{2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (c (B d (8 c d-9 b e)+A e (2 c d-b e)) \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 d e^3 (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt{d+e x}}-\frac{2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt{b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac{2 \sqrt{-b} \sqrt{c} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{-b} \sqrt{c} (B d (8 c d-9 b e)+A e (2 c d-b e)) \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 d e^3 (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.48216, size = 491, normalized size = 0.99 \[ \frac{2 \left (b e x (b+c x) \left ((d+e x)^2 \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2-d (d+e x) (c d-b e) (A e (b e-2 c d)+B d (7 c d-6 b e))\right )-c \sqrt{\frac{b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) (A e (c d-2 b e)+B d (4 c d-3 b e)) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )\right )\right )}{15 b d^2 e^3 \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 - d*(c*d - b*e)*(B*d*(7*c*d - 6*b*e) + A*e*(-2*c*d + b*e)
)*(d + e*x) + (2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e^2))*(d + e*x)^2) -
Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*
e^2))*(b + c*x)*(d + e*x) + I*b*e*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e
^2))*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)*(B*d*(4*c*d - 3*b*e) + A*e*(c*d - 2*b*e))*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)])))/(15*b*d^2*e^3*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))

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Maple [B]  time = 0.066, size = 3831, normalized size = 7.8 \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)^(1/2)/(e*x+d)^(7/2),x)

[Out]

2/15*(-2*A*x^4*b*c^3*d*e^5+2*A*x^4*b^2*c^2*e^6+A*x*b*c^3*d^4*e^2-6*B*x*b^2*c^2*d^4*e^2+4*B*x*b*c^3*d^5*e+3*B*x
^4*b^2*c^2*d*e^5-13*B*x^4*b*c^3*d^2*e^4+3*A*x^3*b^2*c^2*d*e^5-5*A*x^3*b*c^3*d^2*e^4+3*B*x^3*b^3*c*d*e^5-13*B*x
^3*b^2*c^2*d^2*e^4-5*B*x^3*b*c^3*d^3*e^3+5*A*x^2*b^3*c*d*e^5-7*A*x^2*b^2*c^2*d^2*e^4+7*A*x^2*b*c^3*d^3*e^3-13*
B*x^2*b^2*c^2*d^3*e^3+3*B*x^2*b*c^3*d^4*e^2+A*x*b^2*c^2*d^3*e^3+2*A*x^4*c^4*d^2*e^4+8*B*x^4*c^4*d^3*e^3+2*A*x^
3*b^3*c*e^6+6*A*x^3*c^4*d^3*e^3+9*B*x^3*c^4*d^4*e^2+A*x^2*c^4*d^4*e^2+4*B*x^2*c^4*d^5*e-8*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)+9*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)-17*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)+8*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)-8*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)+8*A*El
lipticE(((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)-4*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)+2*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)-6*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)+4*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)-32*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)+42*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)-16*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)+18*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)-34*B*Ellipt
icF(((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)+16*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)-4*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)+4*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)-2*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
)+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)-3*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)+2*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)-16*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)+21*B*Elli
pticE(((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)+3*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)+4*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)+6*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)-4*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)+4*A*Ellip
ticE(((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)-2*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)+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)-3*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)+2*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)-16*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)+21*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)+9*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)-17*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)+2*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)+2*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)+3
*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)-8*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)+8*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))*(x*(c*x+b))^(1/2)/c/(b*e-c*d)^2/d^2/(c*x+b)/x/e^3/(e*x+d)
^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x}{\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)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x)*(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{\sqrt{c x^{2} + b x}{\left (B x + A\right )} \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)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*(B*x + A)*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{\sqrt{x \left (b + c x\right )} \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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(x*(b + c*x))*(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{\sqrt{c x^{2} + b x}{\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)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

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