### 3.405 $$\int \frac{(b x+c x^2)^{5/2}}{(d+e x)^{11/2}} \, dx$$

Optimal. Leaf size=570 $-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{63 d e^6 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4-320 b c^3 d^3 e+160 c^4 d^4\right )+c d^2 \left (111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d^2 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}$

[Out]

(-2*(c*d^2*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3) + e*(160*c^4*d^4 - 320*b*c^3*d^3*e + 17
1*b^2*c^2*d^2*e^2 - 11*b^3*c*d*e^3 - 2*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(63*d^2*e^5*(c*d - b*e)^2*(d + e*x)^(3/2
)) - (2*(d*(16*c^2*d^2 - 11*b*c*d*e - 2*b^2*e^2) + e*(26*c^2*d^2 - 26*b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3
/2))/(63*d*e^3*(c*d - b*e)*(d + e*x)^(7/2)) - (2*(b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (4*Sqrt[-b]*Sqrt
[c]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d^2*e^6*(c*d - b*e)^2*Sqrt[1 + (
e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d*e^6*(c*d
- b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.679917, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.348, Rules used = {732, 810, 843, 715, 112, 110, 117, 116} $-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4-320 b c^3 d^3 e+160 c^4 d^4\right )+c d^2 \left (111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d e^6 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d^2 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(c*d^2*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3) + e*(160*c^4*d^4 - 320*b*c^3*d^3*e + 17
1*b^2*c^2*d^2*e^2 - 11*b^3*c*d*e^3 - 2*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(63*d^2*e^5*(c*d - b*e)^2*(d + e*x)^(3/2
)) - (2*(d*(16*c^2*d^2 - 11*b*c*d*e - 2*b^2*e^2) + e*(26*c^2*d^2 - 26*b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3
/2))/(63*d*e^3*(c*d - b*e)*(d + e*x)^(7/2)) - (2*(b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (4*Sqrt[-b]*Sqrt
[c]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d^2*e^6*(c*d - b*e)^2*Sqrt[1 + (
e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d*e^6*(c*d
- b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

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 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{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{5 \int \frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e}\\ &=-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \int \frac{\left (-\frac{1}{2} b \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )-\frac{1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{(d+e x)^{5/2}} \, dx}{21 d e^3 (c d-b e)}\\ &=-\frac{2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{4 \int \frac{\frac{1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac{1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{63 d^2 e^5 (c d-b e)^2}\\ &=-\frac{2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{63 d e^6 (c d-b e)}+\frac{\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{63 d^2 e^6 (c d-b e)^2}\\ &=-\frac{2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{63 d e^6 (c d-b e) \sqrt{b x+c x^2}}+\frac{\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{63 d^2 e^6 (c d-b e)^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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}{63 d^2 e^6 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\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}{63 d e^6 (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt{b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{4 \sqrt{-b} \sqrt{c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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 )}{63 d^2 e^6 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} \sqrt{c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\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 )}{63 d e^6 (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.5541, size = 610, normalized size = 1.07 $-\frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (d^2 (d+e x)^2 \left (15 b^2 e^2-88 b c d e+88 c^2 d^2\right ) (c d-b e)^2-19 d^3 (d+e x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) (c d-b e)^2-d (d+e x)^3 \left (63 b^2 c d e^2-b^3 e^3-183 b c^2 d^2 e+122 c^3 d^3\right ) (c d-b e)+(d+e x)^4 \left (207 b^2 c^2 d^2 e^2-14 b^3 c d e^3-2 b^4 e^4-386 b c^3 d^3 e+193 c^4 d^4\right )+7 d^4 (c d-b e)^4\right )-c \sqrt{\frac{b}{c}} (d+e x)^4 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (159 b^2 c^2 d^2 e^2-13 b^3 c d e^3-2 b^4 e^4-272 b c^3 d^3 e+128 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4+256 b c^3 d^3 e-128 c^4 d^4\right )\right )\right )}{63 b d^2 e^6 x^3 (b+c x)^3 (d+e x)^{9/2} (c d-b e)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(7*d^4*(c*d - b*e)^4 - 19*d^3*(c*d - b*e)^2*(2*c^2*d^2 - 3*b*c*d*e +
b^2*e^2)*(d + e*x) + d^2*(c*d - b*e)^2*(88*c^2*d^2 - 88*b*c*d*e + 15*b^2*e^2)*(d + e*x)^2 - d*(c*d - b*e)*(122
*c^3*d^3 - 183*b*c^2*d^2*e + 63*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^3 + (193*c^4*d^4 - 386*b*c^3*d^3*e + 207*b^2*
c^2*d^2*e^2 - 14*b^3*c*d*e^3 - 2*b^4*e^4)*(d + e*x)^4) - Sqrt[b/c]*c*(d + e*x)^4*(-2*Sqrt[b/c]*(-128*c^4*d^4 +
256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*(b + c*x)*(d + e*x) + (2*I)*b*e*(128*c^4*d^4
- 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*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*(128*c^4*d^4 - 272*b*c^3*d^3*e + 159*b^2*c^2*d^
2*e^2 - 13*b^3*c*d*e^3 - 2*b^4*e^4)*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)])))/(63*b*d^2*e^6*(c*d - b*e)^2*x^3*(b + c*x)^3*(d + e*x)^(9/2))

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Maple [B]  time = 0.337, size = 5005, normalized size = 8.8 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x)

[Out]

result too large to display

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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{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(e^6*x^6 + 6*d*e^5*x^5 + 15*d^2*e^4*x
^4 + 20*d^3*e^3*x^3 + 15*d^4*e^2*x^2 + 6*d^5*e*x + d^6), 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((c*x**2+b*x)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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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{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)