\(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^{5/2}} \, dx\) [424]

Optimal result

Integrand size = 23, antiderivative size = 344 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (c d-b e) \sqrt {b x+c x^2}}-\frac {16 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (4 c d-3 b e) (4 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(b*d+(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/b^2/(c*x^2+b*x)^(3/2)+2/3*(b*(-5*b*e+8*c*d)*(-b*e+c*d)+8*c*(-b*e+c*d)*
(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/b^4/(-b*e+c*d)/(c*x^2+b*x)^(1/2)-16/3*(-b*e+2*c*d)*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)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^
(1/2)+2/3*(-3*b*e+4*c*d)*(-b*e+4*c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(1+c*x/b)^
(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 344, 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 = {752, 836, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 c d-3 b e) (4 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {16 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt {b x+c x^2} (c d-b e)}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*(8*c*d - 5*b*e)*(
c*d - b*e) + 8*c*(c*d - b*e)*(2*c*d - b*e)*x))/(3*b^4*(c*d - b*e)*Sqrt[b*x + c*x^2]) - (16*Sqrt[c]*(2*c*d - b*
e)*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)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(4*c*d - 3*b*e)*(4*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[c]*Sqrt[d + e*x]*S
qrt[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 752

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 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 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} d (8 c d-5 b e)+\frac {3}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (c d-b e) \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e (8 c d-3 b e) (c d-b e)-2 c d e (c d-b e) (2 c d-b e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)} \\ & = -\frac {2 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (c d-b e) \sqrt {b x+c x^2}}-\frac {(8 c (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4}+\frac {((4 c d-3 b e) (4 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4} \\ & = -\frac {2 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (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 {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 \sqrt {b x+c x^2}}+\frac {\left ((4 c d-3 b e) (4 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 {2 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (8 c (2 c d-b e) \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 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left ((4 c d-3 b e) (4 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 \sqrt {d+e x} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} (b (8 c d-5 b e) (c d-b e)+8 c (c d-b e) (2 c d-b e) x)}{3 b^4 (c d-b e) \sqrt {b x+c x^2}}-\frac {16 \sqrt {c} (2 c d-b e) \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} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (4 c d-3 b e) (4 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 {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.49 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-8 (2 c d-b e) x (b+c x) (d+e x)+\frac {(d+e x) \left (16 c^3 d x^3+b^2 c x (6 d-13 e x)-8 b c^2 x^2 (-3 d+e x)-b^3 (d+4 e x)\right )}{b+c x}+8 i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} c e (8 c d-5 b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{3 b^4 x \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]

[Out]

(2*(-8*(2*c*d - b*e)*x*(b + c*x)*(d + e*x) + ((d + e*x)*(16*c^3*d*x^3 + b^2*c*x*(6*d - 13*e*x) - 8*b*c^2*x^2*(
-3*d + e*x) - b^3*(d + 4*e*x)))/(b + c*x) + (8*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e
*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*c*e*(8*c*d - 5*b*e)*Sqrt[1 + b
/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*b^4*x*Sqrt[x*(b +
c*x)]*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.26 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.65

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {8 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e -2 c d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} c \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \left (c e \,x^{2}+c d x \right ) \left (b e -2 c d \right )}{3 b^{4} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {c d e}{3 b^{3}}-\frac {e \left (b e -c d \right )}{3 b^{3}}+\frac {4 \left (b e -c d \right ) \left (b e -2 c d \right )}{3 b^{4}}+\frac {4 c d \left (b e -2 c d \right )}{3 b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {16 e \left (b e -2 c d \right ) \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{3 b^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(569\)
default	\(\text {Expression too large to display}\)	\(1099\)

[In]

int((e*x+d)^(3/2)/(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*d/b^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/x^
2-8/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4*(b*e-2*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3*(b*e-c*d)/b^3/c*(c*e*x
^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(1/c*b+x)^2-8/3*(c*e*x^2+c*d*x)/b^4*(b*e-2*c*d)/((1/c*b+x)*(c*e*x^2+c*d*x))^(1
/2)+2*(-1/3/b^3*c*d*e-1/3*e*(b*e-c*d)/b^3+4/3*(b*e-c*d)*(b*e-2*c*d)/b^4+4/3*c*d/b^4*(b*e-2*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))+16/3*e*(b*e-2*c*d)/b^3*((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)*EllipticE(((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 = 0.10 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{4} d^{2} - 16 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (16 \, b c^{3} d^{2} - 16 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left ({\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d e - b^{2} c^{2} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d e - b^{3} c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{3} c d e - 8 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{3} - {\left (24 \, b c^{3} d e - 13 \, b^{2} c^{2} e^{2}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{2} d e - 2 \, b^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{3} e x^{4} + 2 \, b^{5} c^{2} e x^{3} + b^{6} c e x^{2}\right )}} \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/9*(((16*c^4*d^2 - 16*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 16*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*
b^2*c^2*d^2 - 16*b^3*c*d*e + b^4*e^2)*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)) + 24*((2*c^4*d*e - b*c^3*e^2)*x^4 + 2*(2*b*c^3*d*e - b^2*c^2*e^2)*x^3 + (2*b^2*c^2*d*e - b^3*c*e^2)*x^2)*s
qrt(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*d*e
- 8*(2*c^4*d*e - b*c^3*e^2)*x^3 - (24*b*c^3*d*e - 13*b^2*c^2*e^2)*x^2 - 2*(3*b^2*c^2*d*e - 2*b^3*c*e^2)*x)*sqr
t(c*x^2 + b*x)*sqrt(e*x + d))/(b^4*c^3*e*x^4 + 2*b^5*c^2*e*x^3 + b^6*c*e*x^2)

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d + e*x)**(3/2)/(x*(b + c*x))**(5/2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^(3/2)/(b*x + c*x^2)^(5/2), x)