### 3.414 $$\int \frac{(d+e x)^{7/2}}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=395 $-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 c^2}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c}$

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (4*e*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2
)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(b^2*c)
+ (2*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcS
in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*
d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*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)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.513945, antiderivative size = 395, 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 = {738, 832, 843, 715, 112, 110, 117, 116} $\frac{4 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 c^2}-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (4*e*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2
)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(b^2*c)
+ (2*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcS
in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*
d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*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)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 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{(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}-\frac{2 \int \frac{(d+e x)^{3/2} \left (-\frac{5}{2} b d e-\frac{5}{2} e (2 c d-b e) x\right )}{\sqrt{b x+c x^2}} \, dx}{b^2}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}-\frac{4 \int \frac{\sqrt{d+e x} \left (-\frac{5}{4} b d e (3 c d+b e)-\frac{5}{2} e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{5 b^2 c}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}-\frac{8 \int \frac{-\frac{5}{8} b d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )-\frac{5}{8} e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 b^2 c^2}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}-\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 c^2}+\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^2 c^2}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}-\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{3 b^2 c^2 \sqrt{b x+c x^2}}+\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 c^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}+\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 c^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{3 b^2 c^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}+\frac{4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{3 b^2 c^2}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{b^2 c}+\frac{2 (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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 )}{3 (-b)^{3/2} c^{5/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{4 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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 )}{3 (-b)^{3/2} c^{5/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.24455, size = 360, normalized size = 0.91 $-\frac{2 \left (b (d+e x) \left (-b^2 e^3 x (b+c x)+3 c^2 d^3 (b+c x)+3 x (c d-b e)^3\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 c d e^2-8 b^3 e^3-18 b c^2 d^2 e+3 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{3 b^3 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(b*(d + e*x)*(3*(c*d - b*e)^3*x + 3*c^2*d^3*(b + c*x) - b^2*e^3*x*(b + c*x)) - Sqrt[b/c]*(Sqrt[b/c]*(6*c^3
*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19
*b^2*c*d*e^2 - 8*b^3*e^3)*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*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*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)])))/(3*b^3*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x
])

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Maple [B]  time = 0.361, size = 863, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(4*((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^3-10*((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+12*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*E
llipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e-6*((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^4+8*((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^4-27*((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+
28*((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^3*c^2*d^2*e^2-15*((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+6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^4+x^3*b^2*c^3*e^4+4*x^2*b^3*c^2*e^4-8*x^2*b^2*c^3*d*e^3
+9*x^2*b*c^4*d^2*e^2-6*x^2*c^5*d^3*e+4*x*b^3*c^2*d*e^3-9*x*b^2*c^3*d^2*e^2+6*x*b*c^4*d^3*e-6*x*c^5*d^4-3*b*c^4
*d^4)/x*(x*(c*x+b))^(1/2)/(c*x+b)/c^4/b^2/(e*x+d)^(1/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^2*x^4 + 2*b*c*x^3 + b^2*
x^2), 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((e*x+d)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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