\(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^{5/2}} \, dx\) [422]

Optimal result

Integrand size = 23, antiderivative size = 383 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(e*x+d)^(5/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^(3/2)+2/3*(b*c*d^2*(-9*b*e+8*c*d)+(-b*e+2*c*d)*(-b^2*e
^2-8*b*c*d*e+8*c^2*d^2)*x)*(e*x+d)^(1/2)/b^4/c/(c*x^2+b*x)^(1/2)-4/3*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^
2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/c^(3
/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*d*(-b*e+c*d)*(-b^2*e^2-16*b*c*d*e+16*c^2*d^2)*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^(3/2)/(e*x+d)^(1/2)/(c*x^
2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 383, 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, 832, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {4 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (x (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right )+b c d^2 (8 c d-9 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}} \]

[In]

Int[(d + e*x)^(7/2)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*c*d^2*(8*c*d -
9*b*e) + (2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - b^2*e^2)*x))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (4*(2*c*d - b*e)*(4
*c^2*d^2 - 4*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])/Sqr
t[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*c^2*d^
2 - 16*b*c*d*e - b^2*e^2)*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)*c^(3/2)*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 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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (8 c d-9 b e)-\frac {1}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 (d+e x)^{5/2} (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} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\frac {1}{4} b d e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )+\frac {1}{2} e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c} \\ & = -\frac {2 (d+e x)^{5/2} (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} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c}-\frac {\left (2 (2 c d-b e) \left (4 c^2 d^2-4 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 c} \\ & = -\frac {2 (d+e x)^{5/2} (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} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \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 {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c \sqrt {b x+c x^2}}-\frac {\left (2 (2 c d-b e) \left (4 c^2 d^2-4 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 c \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} (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} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {\left (2 (2 c d-b e) \left (4 c^2 d^2-4 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 c \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \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 {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 c \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} (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} \left (b c d^2 (8 c d-9 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 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} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \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 {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} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.35 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b (c d-b e)^3 x^2+2 (c d-b e)^2 (4 c d+b e) x^2 (b+c x)-b c d^3 (b+c x)^2+2 c d^2 (4 c d-5 b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} x (b+c x) \left (2 \sqrt {\frac {b}{c}} \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (b+c x) (d+e x)+2 i b e \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^3 d^3-13 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 c (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^(5/2),x]

[Out]

(2*(b*(d + e*x)*(b*(c*d - b*e)^3*x^2 + 2*(c*d - b*e)^2*(4*c*d + b*e)*x^2*(b + c*x) - b*c*d^3*(b + c*x)^2 + 2*c
*d^2*(4*c*d - 5*b*e)*x*(b + c*x)^2) - Sqrt[b/c]*x*(b + c*x)*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c
*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*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^3*d^3
- 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*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*c*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(791\) vs. \(2(329)=658\).

Time = 2.34 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.07

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {4 \left (c e \,x^{2}+b e x +c d x +b d \right ) d^{2} \left (5 b e -4 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^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} c^{3} \left (\frac {b}{c}+x \right )^{2}}+\frac {4 \left (c e \,x^{2}+c d x \right ) \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -7 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right )}{3 c^{2} b^{4} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (\frac {e^{4}}{c^{2}}-\frac {d^{3} c e}{3 b^{3}}-\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) e}{3 c^{2} b^{3}}-\frac {2 \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -7 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) \left (b e -c d \right )}{3 c^{2} b^{4}}-\frac {2 d \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -7 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right )}{3 c \,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 {2 \left (\frac {2 c \,d^{2} e \left (5 b e -4 c d \right )}{3 b^{4}}-\frac {2 \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -7 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) e}{3 c \,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}}\, \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}}\)	\(792\)
default	\(\text {Expression too large to display}\)	\(1687\)

[In]

int((e*x+d)^(7/2)/(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^3/b^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/
x^2-4/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4*d^2*(5*b*e-4*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3*(b^3*e^3-3*b^2
*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/b^3/c^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(1/c*b+x)^2+4/3*(c*e*x^2+c*d*x)*
(b^3*e^3+2*b^2*c*d*e^2-7*b*c^2*d^2*e+4*c^3*d^3)/c^2/b^4/((1/c*b+x)*(c*e*x^2+c*d*x))^(1/2)+2*(e^4/c^2-1/3*d^3/b
^3*c*e-1/3*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/c^2*e/b^3-2/3*(b^3*e^3+2*b^2*c*d*e^2-7*b*c^2*d^2*e+4*
c^3*d^3)/c^2*(b*e-c*d)/b^4-2/3/c*d*(b^3*e^3+2*b^2*c*d*e^2-7*b*c^2*d^2*e+4*c^3*d^3)/b^4)/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*(2/3*c*d^2*e*(5*b*e-4*c*d)/b^4-2/3*(b^3*e^3+2*b^2*c*d*e^2-7*b*c^2*d^2*e
+4*c^3*d^3)/c*e/b^4)/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 = 0.11 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.13 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{6} d^{4} - 32 \, b c^{5} d^{3} e + 13 \, b^{2} c^{4} d^{2} e^{2} + 3 \, b^{3} c^{3} d e^{3} + 2 \, b^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (16 \, b c^{5} d^{4} - 32 \, b^{2} c^{4} d^{3} e + 13 \, b^{3} c^{3} d^{2} e^{2} + 3 \, b^{4} c^{2} d e^{3} + 2 \, b^{5} c e^{4}\right )} x^{3} + {\left (16 \, b^{2} c^{4} d^{4} - 32 \, b^{3} c^{3} d^{3} e + 13 \, b^{4} c^{2} d^{2} e^{2} + 3 \, b^{5} c d e^{3} + 2 \, b^{6} e^{4}\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 ) + 6 \, {\left ({\left (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{4} + 2 \, {\left (8 \, b c^{5} d^{3} e - 12 \, b^{2} c^{4} d^{2} e^{2} + 2 \, b^{3} c^{3} d e^{3} + b^{4} c^{2} e^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d^{3} e - 12 \, b^{3} c^{3} d^{2} e^{2} + 2 \, b^{4} c^{2} d e^{3} + b^{5} c e^{4}\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^{3} d^{3} e - 2 \, {\left (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{3} - {\left (24 \, b c^{5} d^{3} e - 37 \, b^{2} c^{4} d^{2} e^{2} + 7 \, b^{3} c^{3} d e^{3} + b^{4} c^{2} e^{4}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{4} d^{3} e - 5 \, b^{3} c^{3} d^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{5} e x^{4} + 2 \, b^{5} c^{4} e x^{3} + b^{6} c^{3} e x^{2}\right )}} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/9*(((16*c^6*d^4 - 32*b*c^5*d^3*e + 13*b^2*c^4*d^2*e^2 + 3*b^3*c^3*d*e^3 + 2*b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d
^4 - 32*b^2*c^4*d^3*e + 13*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 + 2*b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 32*b^3*c^3
*d^3*e + 13*b^4*c^2*d^2*e^2 + 3*b^5*c*d*e^3 + 2*b^6*e^4)*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)) + 6*((8*c^6*d^3*e - 12*b*c^5*d^2*e^2 + 2*b^2*c^4*d*e^3 + b^3*c^3*e^4)*x^4 + 2*(8*b*c^5*
d^3*e - 12*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^3 + (8*b^2*c^4*d^3*e - 12*b^3*c^3*d^2*e^2 + 2*b^
4*c^2*d*e^3 + b^5*c*e^4)*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^3*d^3*e - 2*(8*c^6*d^3*e - 12*b*c^5*d^2*e^2 + 2*b^2*c^4*d*e^3 + b^3*c^3*e^4)*x^3 -
(24*b*c^5*d^3*e - 37*b^2*c^4*d^2*e^2 + 7*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^2 - 2*(3*b^2*c^4*d^3*e - 5*b^3*c^3*d^2
*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(b^4*c^5*e*x^4 + 2*b^5*c^4*e*x^3 + b^6*c^3*e*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(7/2)/(b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^(7/2)/(b*x + c*x^2)^(5/2), x)