\(\int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx\) [413]

Optimal result

Integrand size = 23, antiderivative size = 403 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {c} \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {8 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[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)*c^(1/2
)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^3/(-b*e+c*d)^3/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-8/15*(-b*e+2*c*d)*E
llipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2
)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/5*e*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^(5/2)-8/15*e*(
-b*e+2*c*d)*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)-2/15*e*(8*b^2*e^2-23*b*c*d*e+23*c^2*d^2)*(c*x^2+b
*x)^(1/2)/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {758, 848, 857, 729, 113, 111, 118, 117} \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {8 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 e \sqrt {b x+c x^2} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{15 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {8 e \sqrt {b x+c x^2} (2 c d-b e)}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 e \sqrt {b x+c x^2}}{5 d (d+e x)^{5/2} (c d-b e)} \]

[In]

Int[1/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (8*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(15*d^2*(c*
d - b*e)^2*(d + e*x)^(3/2)) - (2*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e
)^3*Sqrt[d + e*x]) + (2*Sqrt[-b]*Sqrt[c]*(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[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) - (8*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^2*(c*d - b*e)^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 758

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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} (-5 c d+4 b e)+\frac {3 c e x}{2}}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx}{5 d (c d-b e)} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {1}{4} \left (15 c^2 d^2-19 b c d e+8 b^2 e^2\right )-c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{15 d^2 (c d-b e)^2} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} c d \left (15 c^2 d^2-11 b c d e+4 b^2 e^2\right )-\frac {1}{8} c 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 d^3 (c d-b e)^3} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {(4 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d^2 (c d-b e)^2}+\frac {\left (c \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 d^3 (c d-b e)^3} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (4 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}{15 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \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 d^3 (c d-b e)^3 \sqrt {b x+c x^2}} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (c \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 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (4 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}{15 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 e \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {c} \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 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {8 \sqrt {-b} \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 )}{15 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.12 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 \left (b e x (b+c x) \left (3 d^2 (c d-b e)^2+4 d (c d-b e) (2 c d-b e) (d+e x)+\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) (d+e x)^2\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) (b+c x) (d+e x)+i b e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \left (15 c^3 d^3-34 b c^2 d^2 e+27 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b d^3 (c d-b e)^3 \sqrt {x (b+c x)} (d+e x)^{5/2}} \]

[In]

Integrate[1/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*(b*e*x*(b + c*x)*(3*d^2*(c*d - b*e)^2 + 4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x) + (23*c^2*d^2 - 23*b*c*d*e
+ 8*b^2*e^2)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x
)*(d + e*x) + I*b*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^(3/2)*Elliptic
E[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*(15*c^3*d^3 - 34*b*c^2*d^2*e + 27*b^2*c*d*e^2 - 8*b^3*e^3)*Sq
rt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*d^3*(c
*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 d \,e^{2} \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {8 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e \,d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (8 b^{2} e^{2}-23 b c d e +23 c^{2} d^{2}\right )}{15 d^{3} \left (b e -c d \right )^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {4 c \left (b e -2 c d \right )}{15 d^{2} \left (b e -c d \right )^{2}}+\frac {8 b^{2} e^{2}-23 b c d e +23 c^{2} d^{2}}{15 \left (b e -c d \right )^{2} d^{3}}-\frac {b e \left (8 b^{2} e^{2}-23 b c d e +23 c^{2} d^{2}\right )}{15 d^{3} \left (b e -c d \right )^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 e \left (8 b^{2} e^{2}-23 b c d e +23 c^{2} d^{2}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{15 d^{3} \left (b e -c d \right )^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(639\)
default	\(\text {Expression too large to display}\)	\(1912\)

[In]

int(1/(e*x+d)^(7/2)/(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/d/e^2/(b*e-c*d)*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)
^(1/2)/(x+d/e)^3+8/15*(b*e-2*c*d)/e/d^2/(b*e-c*d)^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+2/15*(c*e*
x^2+b*e*x)/d^3/(b*e-c*d)^3*(8*b^2*e^2-23*b*c*d*e+23*c^2*d^2)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(4/15*c*(b*e-2*
c*d)/d^2/(b*e-c*d)^2+1/15/(b*e-c*d)^2*(8*b^2*e^2-23*b*c*d*e+23*c^2*d^2)/d^3-1/15*b*e/d^3/(b*e-c*d)^3*(8*b^2*e^
2-23*b*c*d*e+23*c^2*d^2))/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))-2/15*e*(8*b^2*e^2-23*b*c*
d*e+23*c^2*d^2)/d^3/(b*e-c*d)^3*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*Ell
ipticF(((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 = 0.20 (sec) , antiderivative size = 950, normalized size of antiderivative = 2.36 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (22 \, c^{3} d^{6} - 33 \, b c^{2} d^{5} e + 27 \, b^{2} c d^{4} e^{2} - 8 \, b^{3} d^{3} e^{3} + {\left (22 \, c^{3} d^{3} e^{3} - 33 \, b c^{2} d^{2} e^{4} + 27 \, b^{2} c d e^{5} - 8 \, b^{3} e^{6}\right )} x^{3} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 33 \, b c^{2} d^{3} e^{3} + 27 \, b^{2} c d^{2} e^{4} - 8 \, b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (22 \, c^{3} d^{5} e - 33 \, b c^{2} d^{4} e^{2} + 27 \, b^{2} c d^{3} e^{3} - 8 \, b^{3} d^{2} e^{4}\right )} x\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^{5} e - 23 \, b c^{2} d^{4} e^{2} + 8 \, b^{2} c d^{3} e^{3} + {\left (23 \, c^{3} d^{2} e^{4} - 23 \, b c^{2} d e^{5} + 8 \, b^{2} c e^{6}\right )} x^{3} + 3 \, {\left (23 \, c^{3} d^{3} e^{3} - 23 \, b c^{2} d^{2} e^{4} + 8 \, b^{2} c d e^{5}\right )} x^{2} + 3 \, {\left (23 \, c^{3} d^{4} e^{2} - 23 \, b c^{2} d^{3} e^{3} + 8 \, b^{2} c d^{2} e^{4}\right )} x\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 (34 \, c^{3} d^{4} e^{2} - 41 \, b c^{2} d^{3} e^{3} + 15 \, b^{2} c d^{2} e^{4} + {\left (23 \, c^{3} d^{2} e^{4} - 23 \, b c^{2} d e^{5} + 8 \, b^{2} c e^{6}\right )} x^{2} + 2 \, {\left (27 \, c^{3} d^{3} e^{3} - 29 \, b c^{2} d^{2} e^{4} + 10 \, b^{2} c d e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, {\left (c^{4} d^{9} e - 3 \, b c^{3} d^{8} e^{2} + 3 \, b^{2} c^{2} d^{7} e^{3} - b^{3} c d^{6} e^{4} + {\left (c^{4} d^{6} e^{4} - 3 \, b c^{3} d^{5} e^{5} + 3 \, b^{2} c^{2} d^{4} e^{6} - b^{3} c d^{3} e^{7}\right )} x^{3} + 3 \, {\left (c^{4} d^{7} e^{3} - 3 \, b c^{3} d^{6} e^{4} + 3 \, b^{2} c^{2} d^{5} e^{5} - b^{3} c d^{4} e^{6}\right )} x^{2} + 3 \, {\left (c^{4} d^{8} e^{2} - 3 \, b c^{3} d^{7} e^{3} + 3 \, b^{2} c^{2} d^{6} e^{4} - b^{3} c d^{5} e^{5}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

2/45*((22*c^3*d^6 - 33*b*c^2*d^5*e + 27*b^2*c*d^4*e^2 - 8*b^3*d^3*e^3 + (22*c^3*d^3*e^3 - 33*b*c^2*d^2*e^4 + 2
7*b^2*c*d*e^5 - 8*b^3*e^6)*x^3 + 3*(22*c^3*d^4*e^2 - 33*b*c^2*d^3*e^3 + 27*b^2*c*d^2*e^4 - 8*b^3*d*e^5)*x^2 +
3*(22*c^3*d^5*e - 33*b*c^2*d^4*e^2 + 27*b^2*c*d^3*e^3 - 8*b^3*d^2*e^4)*x)*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^5*e - 23*b*c^2*d^4*e^2 + 8*b^2*c*d^3*e^3 + (23*c^3*d^2*e^4 -
23*b*c^2*d*e^5 + 8*b^2*c*e^6)*x^3 + 3*(23*c^3*d^3*e^3 - 23*b*c^2*d^2*e^4 + 8*b^2*c*d*e^5)*x^2 + 3*(23*c^3*d^4*
e^2 - 23*b*c^2*d^3*e^3 + 8*b^2*c*d^2*e^4)*x)*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*(34*c^3*d^4*e^2 - 41*b*c^2*d^3*e^3 + 15*b^2*c*d^2*e^4 + (23*c^3*d^2*e^4 - 2
3*b*c^2*d*e^5 + 8*b^2*c*e^6)*x^2 + 2*(27*c^3*d^3*e^3 - 29*b*c^2*d^2*e^4 + 10*b^2*c*d*e^5)*x)*sqrt(c*x^2 + b*x)
*sqrt(e*x + d))/(c^4*d^9*e - 3*b*c^3*d^8*e^2 + 3*b^2*c^2*d^7*e^3 - b^3*c*d^6*e^4 + (c^4*d^6*e^4 - 3*b*c^3*d^5*
e^5 + 3*b^2*c^2*d^4*e^6 - b^3*c*d^3*e^7)*x^3 + 3*(c^4*d^7*e^3 - 3*b*c^3*d^6*e^4 + 3*b^2*c^2*d^5*e^5 - b^3*c*d^
4*e^6)*x^2 + 3*(c^4*d^8*e^2 - 3*b*c^3*d^7*e^3 + 3*b^2*c^2*d^6*e^4 - b^3*c*d^5*e^5)*x)

Sympy [F]

\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(sqrt(x*(b + c*x))*(d + e*x)**(7/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)

Giac [F]

\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^(7/2)),x)

[Out]

int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^(7/2)), x)