### 3.404 $$\int \frac{(b x+c x^2)^{5/2}}{(d+e x)^{9/2}} \, dx$$

Optimal. Leaf size=474 $\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{21 e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{21 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac{2 c \sqrt{b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{21 d e^5 \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 d e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}$

[Out]

(2*c*(d*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2) + e*(32*c^2*d^2 - 32*b*c*d*e + 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2
])/(21*d*e^5*(c*d - b*e)*Sqrt[d + e*x]) - (2*(c*d^2*(16*c*d - 13*b*e) + e*(22*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2
)*x)*(b*x + c*x^2)^(3/2))/(21*d*e^3*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2
)) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*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)])/(21*d*e^6*(c*d - b*e)*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) + (4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 27*b^2*e^2)*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)])/(21*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*
x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.561803, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.391, Rules used = {732, 810, 812, 843, 715, 112, 110, 117, 116} $-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{21 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac{2 c \sqrt{b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{21 d e^5 \sqrt{d+e x} (c d-b e)}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 d e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(2*c*(d*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2) + e*(32*c^2*d^2 - 32*b*c*d*e + 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2
])/(21*d*e^5*(c*d - b*e)*Sqrt[d + e*x]) - (2*(c*d^2*(16*c*d - 13*b*e) + e*(22*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2
)*x)*(b*x + c*x^2)^(3/2))/(21*d*e^3*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2
)) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*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)])/(21*d*e^6*(c*d - b*e)*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) + (4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 27*b^2*e^2)*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)])/(21*e^6*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 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{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{5 \int \frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{2 \int \frac{\left (-\frac{1}{2} b c d (16 c d-13 b e)-\frac{1}{2} c \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{(d+e x)^{3/2}} \, dx}{7 d e^3 (c d-b e)}\\ &=\frac{2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{21 d e^5 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{4 \int \frac{-\frac{1}{4} b c d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )-\frac{1}{4} c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{21 d e^5 (c d-b e)}\\ &=\frac{2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{21 d e^5 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{21 d e^6 (c d-b e)}+\frac{\left (2 c \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{21 e^6}\\ &=\frac{2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{21 d e^5 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{21 d e^6 (c d-b e) \sqrt{b x+c x^2}}+\frac{\left (2 c \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt{b x+c x^2}}\\ &=\frac{2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{21 d e^5 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\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}{21 d e^6 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 c \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{21 d e^5 (c d-b e) \sqrt{d+e x}}-\frac{2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{2 \sqrt{-b} \sqrt{c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\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 )}{21 d e^6 (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \sqrt{-b} \sqrt{c} \left (128 c^2 d^2-128 b c d e+27 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 )}{21 e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.5287, size = 500, normalized size = 1.05 $-\frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (-b^2 c d e^2 \left (169 d^2 e x+51 d^3+194 d e^2 x^2+85 e^3 x^3\right )+3 b^3 e^6 x^3+b c^2 d e \left (649 d^2 e^2 x^2+576 d^3 e x+176 d^4+265 d e^3 x^3+7 e^4 x^4\right )-c^3 d^2 \left (464 d^2 e^2 x^2+416 d^3 e x+128 d^4+186 d e^3 x^3+7 e^4 x^4\right )\right )+c \sqrt{\frac{b}{c}} (d+e x)^3 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (83 b^2 c d e^2-3 b^3 e^3-208 b c^2 d^2 e+128 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (134 b^2 c d e^2-3 b^3 e^3-384 b c^2 d^2 e+256 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (134 b^2 c d e^2-3 b^3 e^3-384 b c^2 d^2 e+256 c^3 d^3\right )\right )\right )}{21 b d e^6 x^3 (b+c x)^3 (d+e x)^{7/2} (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(3*b^3*e^6*x^3 - b^2*c*d*e^2*(51*d^3 + 169*d^2*e*x + 194*d*e^2*x^2 +
85*e^3*x^3) - c^3*d^2*(128*d^4 + 416*d^3*e*x + 464*d^2*e^2*x^2 + 186*d*e^3*x^3 + 7*e^4*x^4) + b*c^2*d*e*(176*d
^4 + 576*d^3*e*x + 649*d^2*e^2*x^2 + 265*d*e^3*x^3 + 7*e^4*x^4)) + Sqrt[b/c]*c*(d + e*x)^3*(Sqrt[b/c]*(256*c^3
*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(256*c^3*d^3 - 384*b*c^2*d^2
*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)] - I*b*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sq
rt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(21*b*d*e^6*(c*d - b*e)*x^3*(b
+ c*x)^3*(d + e*x)^(7/2))

________________________________________________________________________________________

Maple [B]  time = 0.307, size = 3284, normalized size = 6.9 \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)^(9/2),x)

[Out]

2/21*(x*(c*x+b))^(1/2)*(-310*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^3*c^2*d^2*e^5*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-137*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3
*b^4*c*d*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+518*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*x^3*b^3*c^2*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-256*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b*c^4*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)-640*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^2*c^3*d^3*e^4*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-411*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*c*d
^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+1554*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*x^2*b^3*c^2*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-1920*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^3*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*
(-c*x/b)^(1/2)+768*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^4*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+162*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*c*d^2*e
^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+256*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x^3*b*c^4*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-930*EllipticF(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^2*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)+1536*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^3*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-768*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^4*d^5*e^2*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-411*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*x*b^4*c*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+1554*EllipticE(((c*x+b)/b)^
(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^2*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-1
920*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^3*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*
d))^(1/2)*(-c*x/b)^(1/2)+768*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d^6*e*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-7*x^6*c^5*d^2*e^5+9*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*x*b^5*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-137*EllipticE(((c*x+b)/b)^(
1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+518*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)-640*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^6*e*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+54*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^4*e^3*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-310*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2
))*b^3*c^2*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+512*EllipticF(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+258*x^5*
b*c^4*d^2*e^5-85*x^4*b^3*c^2*d*e^6+71*x^4*b^2*c^3*d^2*e^5+463*x^4*b*c^4*d^3*e^4-194*x^3*b^3*c^2*d^2*e^5+480*x^
3*b^2*c^3*d^3*e^4+112*x^3*b*c^4*d^4*e^3-169*x^2*b^3*c^2*d^3*e^4+525*x^2*b^2*c^3*d^4*e^3-240*x^2*b*c^4*d^5*e^2-
51*x*b^3*c^2*d^4*e^3+176*x*b^2*c^3*d^5*e^2-128*x*b*c^4*d^6*e-78*x^5*b^2*c^3*d*e^6+162*EllipticF(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*c*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-930*E
llipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^2*d^4*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(
1/2)*(-c*x/b)^(1/2)+1536*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^3*d^5*e^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-768*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d
^6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+54*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*x^3*b^4*c*d*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+512*EllipticF(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^2*c^3*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+7*x^6*b*c^4*d*e^6+3*x^5*b^3*c^2*e^7-186*x^5*c^5*d^3*e^4+3*x^4*b^4*c*e^7-464*x^4*c^5*d^4*e^3-416*x^3*c^5*
d^5*e^2-128*x^2*c^5*d^6*e+3*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^5*e^7*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d^3*e^4*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+256*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*b*c^4*d^7*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-256*EllipticF(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b*c^4*d^7*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+9*EllipticE(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*d*e^6*((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)^(7/2)/d/e^6/c

________________________________________________________________________________________

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{9}{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)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)

________________________________________________________________________________________

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^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, 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)^(9/2),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^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x
^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5), x)

________________________________________________________________________________________

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)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

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{9}{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)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)