\(\int \frac {\sqrt {d+e x}}{(b x+c x^2)^{5/2}} \, dx\) [425]

Optimal result

Integrand size = 23, antiderivative size = 359 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {16 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(2*c*x+b)*(e*x+d)^(1/2)/b^2/(c*x^2+b*x)^(3/2)+2/3*(b*(-b*e+c*d)*(-b*e+8*c*d)+c*(b^2*e^2-16*b*c*d*e+16*c^2
*d^2)*x)*(e*x+d)^(1/2)/b^4/d/(-b*e+c*d)/(c*x^2+b*x)^(1/2)-2/3*(b^2*e^2-16*b*c*d*e+16*c^2*d^2)*EllipticE(c^(1/2
)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/d/(-b*e+c*d)/(1
+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+16/3*(-b*e+2*c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2
)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {750, 836, 857, 729, 113, 111, 118, 117} \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {16 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (c x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (c d-b e) (8 c d-b e)\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)} \]

[In]

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

[Out]

(-2*(b + 2*c*x)*Sqrt[d + e*x])/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*c*d - b*e) + c
*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x))/(3*b^4*d*(c*d - b*e)*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*c^2*d^2 - 16
*b*c*d*e + 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)])/(3*(-b)^(7/2)*d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (16*Sqrt[c]*(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)])/(3*(-b)^(7/2)
*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 113

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 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

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 729

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 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*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
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*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
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {-4 c d+\frac {b e}{2}-3 c e x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {4 \int \frac {\frac {1}{4} b c d e (8 c d-7 b e)+\frac {1}{4} c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (8 c (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}{3 b^4 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+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^4 d (c d-b e) \sqrt {b x+c x^2}} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+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^4 d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (8 c (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}{3 b^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e+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)^{7/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {16 \sqrt {c} (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 )}{3 (-b)^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.43 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b c^2 d (c d-b e) x^2+c^2 d (8 c d-7 b e) x^2 (b+c x)+b d (-c d+b e) (b+c x)^2+(c d-b e) (8 c d-b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} c x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) (b+c x) (d+e x)+i b e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^2 d^2-9 b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d (c d-b e) (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(b*(d + e*x)*(b*c^2*d*(c*d - b*e)*x^2 + c^2*d*(8*c*d - 7*b*e)*x^2*(b + c*x) + b*d*(-(c*d) + b*e)*(b + c*x)^
2 + (c*d - b*e)*(8*c*d - b*e)*x*(b + c*x)^2) - Sqrt[b/c]*c*x*(b + c*x)*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b
^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(16*c^2*d^2 - 16*b*c*d*e + 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*(8*c^2*d^2 - 9*b*c*d*e + b^2*e^2)*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^5*d*(c*d - b*e
)*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e -8 c d \right )}{3 b^{4} d \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \left (c e \,x^{2}+c d x \right ) c \left (7 b e -8 c d \right )}{3 b^{4} \left (b e -c d \right ) \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {c \left (7 b e -8 c d \right )}{3 b^{4}}-\frac {c^{2} d \left (7 b e -8 c d \right )}{3 b^{4} \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c e \left (b e -8 c d \right )}{3 b^{4} d}-\frac {c^{2} e \left (7 b e -8 c d \right )}{3 b^{4} \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(596\)
default	\(\text {Expression too large to display}\)	\(1362\)

[In]

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

[Out]

(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3/b^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/x^2-
2/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4/d*(b*e-8*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)+2/3/b^3*(c*e*x^3+b*e*x^2+c
*d*x^2+b*d*x)^(1/2)/(1/c*b+x)^2+2/3*(c*e*x^2+c*d*x)/b^4/(b*e-c*d)*c*(7*b*e-8*c*d)/((1/c*b+x)*(c*e*x^2+c*d*x))^
(1/2)+2*(-1/3*c*(7*b*e-8*c*d)/b^4-1/3*c^2*d/b^4/(b*e-c*d)*(7*b*e-8*c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-
1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b
/(-1/c*b+d/e))^(1/2))+2*(1/3*c*e*(b*e-8*c*d)/b^4/d-1/3*c^2*e*(7*b*e-8*c*d)/b^4/(b*e-c*d))/c*b*((1/c*b+x)*c/b)^
(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/e)*Elliptic
E(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))
^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{5} d^{3} - 24 \, b c^{4} d^{2} e + 6 \, b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{4} + 2 \, {\left (16 \, b c^{4} d^{3} - 24 \, b^{2} c^{3} d^{2} e + 6 \, b^{3} c^{2} d e^{2} + b^{4} c e^{3}\right )} x^{3} + {\left (16 \, b^{2} c^{3} d^{3} - 24 \, b^{3} c^{2} d^{2} e + 6 \, b^{4} c d e^{2} + b^{5} e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (16 \, c^{5} d^{2} e - 16 \, b c^{4} d e^{2} + b^{2} c^{3} e^{3}\right )} x^{4} + 2 \, {\left (16 \, b c^{4} d^{2} e - 16 \, b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} + {\left (16 \, b^{2} c^{3} d^{2} e - 16 \, b^{3} c^{2} d e^{2} + b^{4} c e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{3} c^{2} d^{2} e - b^{4} c d e^{2} - {\left (16 \, c^{5} d^{2} e - 16 \, b c^{4} d e^{2} + b^{2} c^{3} e^{3}\right )} x^{3} - {\left (24 \, b c^{4} d^{2} e - 25 \, b^{2} c^{3} d e^{2} + 2 \, b^{3} c^{2} e^{3}\right )} x^{2} - {\left (6 \, b^{2} c^{3} d^{2} e - 7 \, b^{3} c^{2} d e^{2} + b^{4} c e^{3}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left ({\left (b^{4} c^{4} d^{2} e - b^{5} c^{3} d e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{2} e - b^{6} c^{2} d e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{2} e - b^{7} c d e^{2}\right )} x^{2}\right )}} \]

[In]

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

[Out]

2/9*(((16*c^5*d^3 - 24*b*c^4*d^2*e + 6*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(16*b*c^4*d^3 - 24*b^2*c^3*d^2*e +
6*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (16*b^2*c^3*d^3 - 24*b^3*c^2*d^2*e + 6*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(c*
e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c
*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*((16*c^5*d^2*e - 16*b*c^4*d*e^2 + b^2*c^3*
e^3)*x^4 + 2*(16*b*c^4*d^2*e - 16*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^3 + (16*b^2*c^3*d^2*e - 16*b^3*c^2*d*e^2 + b^
4*c*e^3)*x^2)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^
2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2
), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)))
- 3*(b^3*c^2*d^2*e - b^4*c*d*e^2 - (16*c^5*d^2*e - 16*b*c^4*d*e^2 + b^2*c^3*e^3)*x^3 - (24*b*c^4*d^2*e - 25*b^
2*c^3*d*e^2 + 2*b^3*c^2*e^3)*x^2 - (6*b^2*c^3*d^2*e - 7*b^3*c^2*d*e^2 + b^4*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(e
*x + d))/((b^4*c^4*d^2*e - b^5*c^3*d*e^2)*x^4 + 2*(b^5*c^3*d^2*e - b^6*c^2*d*e^2)*x^3 + (b^6*c^2*d^2*e - b^7*c
*d*e^2)*x^2)

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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