### 3.407 $$\int \frac{(d+e x)^{5/2}}{\sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=303 $-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c}$

[Out]

(8*e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*e*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*c) +
(2*Sqrt[-b]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(S
qrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*Sqrt[-b]*d*(c*d
- b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.330066, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.348, Rules used = {742, 832, 843, 715, 112, 110, 117, 116} $\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(8*e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*e*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*c) +
(2*Sqrt[-b]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(S
qrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*Sqrt[-b]*d*(c*d
- b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && 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)^{5/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{2 \int \frac{\sqrt{d+e x} \left (\frac{1}{2} d (5 c d-b e)+2 e (2 c d-b e) x\right )}{\sqrt{b x+c x^2}} \, dx}{5 c}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{4 \int \frac{\frac{1}{4} d \left (15 c^2 d^2-11 b c d e+4 b^2 e^2\right )+\frac{1}{4} e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}-\frac{(4 d (c d-b e) (2 c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 c^2}+\frac{\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}-\frac{\left (4 d (c d-b 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 c^2 \sqrt{b x+c x^2}}+\frac{\left (\left (23 c^2 d^2-23 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}{15 c^2 \sqrt{b x+c x^2}}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{\left (\left (23 c^2 d^2-23 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}{15 c^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (4 d (c d-b 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 c^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{2 \sqrt{-b} \left (23 c^2 d^2-23 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 )}{15 c^{5/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{8 \sqrt{-b} d (c d-b 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 c^{5/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.28467, size = 314, normalized size = 1.04 $\frac{2 \sqrt{x} \left (-\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-27 b^2 c d e^2+8 b^3 e^3+34 b c^2 d^2 e-15 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )}{b}+\frac{(b+c x) (d+e x) \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+e \sqrt{x} (b+c x) (d+e x) (-4 b e+11 c d+3 c e x)\right )}{15 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[x]*(((23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) + e*Sqrt[x]*(b + c*x)*(d +
e*x)*(11*c*d - 4*b*e + 3*c*e*x) + I*Sqrt[b/c]*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[
1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (I*Sqrt[b/c]*(-15*c^3*d^3 + 34*b*c^2*d^2
*e - 27*b^2*c*d*e^2 + 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]],
(c*d)/(b*e)])/b))/(15*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [B]  time = 0.272, size = 682, normalized size = 2.3 \begin{align*} -{\frac{2}{15\,x \left ( ce{x}^{2}+bxe+cdx+bd \right ){c}^{4}}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ( 4\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}-12\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e+8\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}+8\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{4}{e}^{3}-31\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}+46\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e-23\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}-3\,{x}^{4}{c}^{4}{e}^{3}+{x}^{3}b{c}^{3}{e}^{3}-14\,{x}^{3}{c}^{4}d{e}^{2}+4\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-10\,{x}^{2}b{c}^{3}d{e}^{2}-11\,{x}^{2}{c}^{4}{d}^{2}e+4\,x{b}^{2}{c}^{2}d{e}^{2}-11\,xb{c}^{3}{d}^{2}e \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-2/15*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(4*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*b^3*c*d*e^2-12*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*b^2*c^2*d^2*e+8*(-c*x/b)^(1/2)*EllipticF(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*b*c^3*d^3+8*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*b^4*e^3-31*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*
b^3*c*d*e^2+46*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*b^2*c^2*d^2*e-23*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*b*c^3*d^3-3*x^4*c^4*e^3+x^3*b*c^3*e^3-14*x^3*c^4*d*e^2+4*x^2*b^2*c^2*e^3
-10*x^2*b*c^3*d*e^2-11*x^2*c^4*d^2*e+4*x*b^2*c^2*d*e^2-11*x*b*c^3*d^2*e)/x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^4

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{5}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((d + e*x)**(5/2)/sqrt(x*(b + c*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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