\(\int \frac {(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^{3/2}} \, dx\) [1272]

Optimal result

Integrand size = 28, antiderivative size = 399 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2*(e*x+d)^(3/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)^(1/2)+2/3*(6*A*c^3*d^2-8*b^3*
B*e^2-3*b*c^2*d*(2*A*e+B*d)+b^2*c*e*(6*A*e+13*B*d))*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)^(3/2)/c^(5/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*d*(-b*e+c*d)*(6*A*c
^2*d+4*b^2*B*e-3*b*c*(A*e+B*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)^(3/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*e*(6*A*c^2*d+4*b^2*B*e-3*b*c*(A*e+B*d))
*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/b^2/c^2

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {832, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right )}{3 b^2 c^2} \]

[In]

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

[Out]

(-2*(d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*Sqrt[b*x + c*x^2]) + (2*e*(6
*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*(6*A*c^3*d^2 - 8*b
^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Ellipti
cE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]
) - (2*d*(c*d - b*e)*(6*A*c^2*d + 4*b^2*B*e - 3*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*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 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 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 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} b (b B+3 A c) d e+\frac {1}{2} e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{b^2 c} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {4 \int \frac {\frac {1}{4} b d e \left (3 A c^2 d-4 b^2 B e+3 b c (2 B d+A e)\right )+\frac {1}{4} e \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c^2}+\frac {\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^2 c^2 \sqrt {b x+c x^2}}+\frac {\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^2 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\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^2 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\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^2 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\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)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\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)^{3/2} 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 = 24.12 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b (d+e x) \left (3 (b B-A c) (c d-b e)^2 x-3 A c^2 d^2 (b+c x)+b^2 B e^2 x (b+c x)\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) (b+c x) (d+e x)+i b e \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (3 A c^2 d+8 b^2 B e-3 b c (3 B d+2 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^3 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(b*(d + e*x)*(3*(b*B - A*c)*(c*d - b*e)^2*x - 3*A*c^2*d^2*(b + c*x) + b^2*B*e^2*x*(b + c*x)) + Sqrt[b/c]*(S
qrt[b/c]*(6*A*c^3*d^2 - 8*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*(b + c*x)*(d + e*x)
+ I*b*e*(6*A*c^3*d^2 - 8*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(13*B*d + 6*A*e))*Sqrt[1 + b/(c*x)]*Sqr
t[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(3*A*c^2*d + 8
*b^2*B*e - 3*b*c*(3*B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)])))/(3*b^3*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. \(801\) vs. \(2(347)=694\).

Time = 1.21 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.01

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) A \,d^{2}}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right )}{b^{2} c^{3} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 B \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{2}}+\frac {2 \left (-\frac {e \left (A b c \,e^{2}-3 A \,c^{2} d e -B \,b^{2} e^{2}+3 B b c d e -3 B \,c^{2} d^{2}\right )}{c^{3}}+\frac {\left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) \left (b e -c d \right )}{c^{3} b^{2}}+\frac {d \left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right )}{c^{2} b^{2}}-\frac {B \,e^{2} b d}{3 c^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\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 (x +\frac {b}{c}\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 {e^{2} \left (A c e -B b e +3 B c d \right )}{c^{2}}+\frac {A c \,d^{2} e}{b^{2}}+\frac {\left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) e}{c^{2} b^{2}}-\frac {2 B \,e^{2} \left (b e +c d \right )}{3 c^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\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 (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\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}}\)	\(802\)
default	\(\text {Expression too large to display}\)	\(1324\)

[In]

int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2*(c*e*x^2+b*e*x+c*d*x+b*d)*A*d^2/b^2/(x*(c*e*x^2+
b*e*x+c*d*x+b*d))^(1/2)-2*(c*e*x^2+c*d*x)*(A*b^2*c*e^2-2*A*b*c^2*d*e+A*c^3*d^2-B*b^3*e^2+2*B*b^2*c*d*e-B*b*c^2
*d^2)/b^2/c^3/((x+b/c)*(c*e*x^2+c*d*x))^(1/2)+2/3*B*e^2/c^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(-e*(A*b*c
*e^2-3*A*c^2*d*e-B*b^2*e^2+3*B*b*c*d*e-3*B*c^2*d^2)/c^3+(A*b^2*c*e^2-2*A*b*c^2*d*e+A*c^3*d^2-B*b^3*e^2+2*B*b^2
*c*d*e-B*b*c^2*d^2)/c^3*(b*e-c*d)/b^2+1/c^2*d*(A*b^2*c*e^2-2*A*b*c^2*d*e+A*c^3*d^2-B*b^3*e^2+2*B*b^2*c*d*e-B*b
*c^2*d^2)/b^2-1/3*B*e^2/c^2*b*d)*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+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(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(1/c^2*e^2*(A*c*e-B*b*e+3*
B*c*d)+A*c*d^2*e/b^2+(A*b^2*c*e^2-2*A*b*c^2*d*e+A*c^3*d^2-B*b^3*e^2+2*B*b^2*c*d*e-B*b*c^2*d^2)/c^2*e/b^2-2/3*B
*e^2/c^2*(b*e+c*d))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+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)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(
1/2),(-b/c/(-b/c+d/e))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (8 \, B b^{2} c^{3} + 9 \, A b c^{4}\right )} d^{2} e - {\left (17 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} e^{3}\right )} x^{2} + {\left (3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + {\left (8 \, B b^{3} c^{2} + 9 \, A b^{2} c^{3}\right )} d^{2} e - {\left (17 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} d e^{2} + 2 \, {\left (4 \, B b^{5} - 3 \, A b^{4} c\right )} e^{3}\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 ({\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e - {\left (13 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} d e^{2} + 2 \, {\left (4 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} e^{3}\right )} x^{2} + {\left (3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} e - {\left (13 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} e^{3}\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 (B b^{2} c^{3} e^{3} x^{2} - 3 \, A b c^{4} d^{2} e + {\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e - 6 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d e^{2} + {\left (4 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{2} c^{5} e x^{2} + b^{3} c^{4} e x\right )}} \]

[In]

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

[Out]

2/9*(((3*(B*b*c^4 - 2*A*c^5)*d^3 + (8*B*b^2*c^3 + 9*A*b*c^4)*d^2*e - (17*B*b^3*c^2 - 9*A*b^2*c^3)*d*e^2 + 2*(4
*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (8*B*b^3*c^2 + 9*A*b^2*c^3)*d^2*e - (17*B*
b^4*c - 9*A*b^3*c^2)*d*e^2 + 2*(4*B*b^5 - 3*A*b^4*c)*e^3)*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*((3*(B*b*c^4 - 2*A*c^5)*d^2*e - (13*B*b^2*c^3 - 6*A*b*c^4)*d*e^2 + 2*(4*B*b^3*c^2 -
3*A*b^2*c^3)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^2*e - (13*B*b^3*c^2 - 6*A*b^2*c^3)*d*e^2 + 2*(4*B*b^4*c -
3*A*b^3*c^2)*e^3)*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*(B*b^2*c^3*e^3*x^2 - 3*A*b*c^4*d^2*e + (3*(B*b*c^4 - 2*A*c^5)*d^2*e - 6*(B*b^2*c^3 - A*b*c^4)*d*e^
2 + (4*B*b^3*c^2 - 3*A*b^2*c^3)*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(b^2*c^5*e*x^2 + b^3*c^4*e*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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