\(\int \frac {(d+e x)^{5/2}}{\sqrt {b x+c x^2}} \, dx\) [407]

Optimal result

Integrand size = 23, antiderivative size = 303 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {b x+c x^2}} \, dx=\frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {2 \sqrt {-b} \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {8 \sqrt {-b} d (c d-b e) (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

2/15*(8*b^2*e^2-23*b*c*d*e+23*c^2*d^2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2
)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/c^(5/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-8/15*d*(-b*e+c*d)*(-b*e+2*c*d)*Ellip
ticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(5/2)/(e
*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/5*e*(e*x+d)^(3/2)*(c*x^2+b*x)^(1/2)/c+8/15*e*(-b*e+2*c*d)*(e*x+d)^(1/2)*(c*x^2
+b*x)^(1/2)/c^2

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 303, 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 = {756, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{5/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {8 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {8 e \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{15 c^2}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c} \]

[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 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 756

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 846

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 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 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] (verified)

Result contains complex when optimal does not.

Time = 18.64 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x} \left (\frac {\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) (b+c x) (d+e x)}{c \sqrt {x}}+e \sqrt {x} (b+c x) (d+e x) (11 c d-4 b e+3 c e x)+i \sqrt {\frac {b}{c}} e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-\frac {i \sqrt {\frac {b}{c}} \left (-15 c^3 d^3+34 b c^2 d^2 e-27 b^2 c d e^2+8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b}\right )}{15 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(249)=498\).

Time = 1.90 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.67

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 e^{2} x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c}+\frac {2 \left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (d^{3}-\frac {\left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) b d}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (3 d^{2} e -\frac {3 e^{2} b d}{5 c}-\frac {2 \left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(505\)
default	\(-\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}-12 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e +8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+8 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} e^{3}-31 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c d \,e^{2}+46 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{2} d^{2} e -23 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{3} d^{3}-3 c^{4} e^{3} x^{4}+b \,c^{3} e^{3} x^{3}-14 c^{4} d \,e^{2} x^{3}+4 b^{2} c^{2} e^{3} x^{2}-10 b \,c^{3} d \,e^{2} x^{2}-11 c^{4} d^{2} e \,x^{2}+4 b^{2} c^{2} d \,e^{2} x -11 b \,c^{3} d^{2} e x \right )}{15 x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{4}}\)	\(682\)

[In]

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

[Out]

(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/5*e^2/c*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2
/3*(3*d*e^2-2/5*e^2/c*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(d^3-1/3*(3*d*e^2-2/5*e^2/c*(
2*b*e+2*c*d))/c/e*b*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*(3*d^2*e-3/5*e^2/c*b*d-2/3
*(3*d*e^2-2/5*e^2/c*(2*b*e+2*c*d))/c/e*(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)*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 = 0.19 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (22 \, c^{3} d^{3} - 33 \, b c^{2} d^{2} e + 27 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 3 \, {\left (23 \, c^{3} d^{2} e - 23 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{3} x + 11 \, c^{3} d e^{2} - 4 \, b c^{2} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, c^{4} e} \]

[In]

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

[Out]

2/45*((22*c^3*d^3 - 33*b*c^2*d^2*e + 27*b^2*c*d*e^2 - 8*b^3*e^3)*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*(23*c^3*d^2*e - 23*b*c^2*d*e^2 + 8*b^2*c*e^3)*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*(3*c^3*e^3*x + 11*c^3*d*e^2 - 4*b*c^2*e^3)*s
qrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^4*e)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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