3.5.20 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^{3/2}} \, dx\) [420]

Optimal. Leaf size=478 \[ -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \sqrt {c} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {754, 848, 857, 729, 113, 111, 118, 117} \begin {gather*} -\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {4 e \sqrt {b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 e \sqrt {b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2]) - (4*e*(3*c^2*d
^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e)*
(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (2*Sqrt[c]*(2
*c*d - b*e)*(3*c^2*d^2 - 3*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)])/(3*(-b)^(3/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
4*Sqrt[c]*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntQuadraticQ[a, b, c, d, e, m, p, x]

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*} \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (3 c d-4 b e)+\frac {3}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {4 \int \frac {-\frac {1}{4} b e \left (3 c^2 d^2-15 b c d e+8 b^2 e^2\right )-\frac {1}{2} c e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} b c d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )-\frac {1}{8} c e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 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^2 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 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}{3 b^2 d^3 (c d-b e)^3 \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 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}{3 b^2 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 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^2 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \sqrt {c} (2 c d-b e) \left (3 c^2 d^2-3 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 )}{3 (-b)^{3/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} \left (3 c^2 d^2-3 b c d e+2 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)^{3/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.38, size = 420, normalized size = 0.88 \begin {gather*} -\frac {2 \left (b \left (b^2 d e^3 (c d-b e) x (b+c x)-5 b^2 e^3 (-2 c d+b e) x (b+c x) (d+e x)+3 c^4 d^3 x (d+e x)^2+3 (c d-b e)^3 (b+c x) (d+e x)^2\right )-\sqrt {\frac {b}{c}} c (d+e x) \left (\sqrt {\frac {b}{c}} \left (6 c^3 d^3-9 b c^2 d^2 e+19 b^2 c d e^2-8 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (6 c^3 d^3-9 b c^2 d^2 e+19 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} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (3 c^3 d^3-18 b c^2 d^2 e+23 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} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{3 b^3 d^3 (c d-b e)^3 \sqrt {x (b+c x)} (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*(b^2*d*e^3*(c*d - b*e)*x*(b + c*x) - 5*b^2*e^3*(-2*c*d + b*e)*x*(b + c*x)*(d + e*x) + 3*c^4*d^3*x*(d +
e*x)^2 + 3*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e
+ 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*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*
(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elliptic
F[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1707\) vs. \(2(420)=840\).
time = 0.48, size = 1708, normalized size = 3.57

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \left (c e \,x^{2}+c d x \right ) c^{3}}{\left (b e -c d \right )^{3} b^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )^{2}}-\frac {10 \left (c e \,x^{2}+b e x \right ) e^{2} \left (b e -2 c d \right )}{3 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 (c e \,x^{2}+b e x +c d x +b d \right )}{d^{3} b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (-\frac {c^{3}}{\left (b e -c d \right )^{2} b^{2}}-\frac {c^{4} d}{\left (b e -c d \right )^{3} b^{2}}-\frac {e^{2} c}{3 \left (b e -c d \right )^{2} d^{2}}-\frac {5 e^{2} \left (b e -2 c d \right )}{3 \left (b e -c d \right )^{2} d^{3}}+\frac {5 b \,e^{3} \left (b e -2 c d \right )}{3 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}}\, \EllipticF \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}+x b d}}+\frac {2 \left (-\frac {c^{4} e}{\left (b e -c d \right )^{3} b^{2}}+\frac {5 e^{3} c \left (b e -2 c d \right )}{3 \left (b e -c d \right )^{3} d^{3}}+\frac {c e}{b^{2} d^{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}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \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 \EllipticF \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}+x b d}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(692\)
default	\(\text {Expression too large to display}\)	\(1708\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(-26*b^3*c^2*d^2*e^3*x+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*e^3-10*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3*e^2+12*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^4*e-27*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*e^3+2
8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*b^3*c^2*d^3*e^2-15*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^4*e+9*b*c^4*d^2*e^3*x^3-7*b^3*c^2*d*e^4*x^2+8*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^5*x-6*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5
+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*b^5*d*e^4+6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b*c^4*d^5-6*x*c^5*d^5+3*b*c^4*d^4*e*x+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^4*x-10*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^3*x+12*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^
2*c^3*d^3*e^2*x-6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b*c^4*d^4*e*x-27*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^4*x+28*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^3*x-15*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e^2*x+6*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*
c^4*d^4*e*x+8*x^2*b^4*c*e^5-3*b*c^4*d^5+8*b^3*c^2*e^5*x^3-6*c^5*d^3*e^2*x^3-12*c^5*d^4*e*x^2-20*b^2*c^3*d^2*e^
3*x^2+15*b*c^4*d^3*e^2*x^2+9*b^2*c^3*d^3*e^2*x-9*b^3*c^2*d^3*e^2+12*x*b^4*c*d*e^4-19*b^2*c^3*d*e^4*x^3+9*b^2*c
^3*d^4*e+3*b^4*c*d^2*e^3)/(e*x+d)^(3/2)/x*(x*(c*x+b))^(1/2)/d^3/c/b^2/(b*e-c*d)^3/(c*x+b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(x*e + d)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.74, size = 1276, normalized size = 2.67 \begin {gather*} \frac {2 \, {\left ({\left (6 \, c^{5} d^{6} x^{2} + 6 \, b c^{4} d^{6} x - 8 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )} e^{6} + {\left (23 \, b^{3} c^{2} d x^{4} + 7 \, b^{4} c d x^{3} - 16 \, b^{5} d x^{2}\right )} e^{5} - {\left (17 \, b^{2} c^{3} d^{2} x^{4} - 29 \, b^{3} c^{2} d^{2} x^{3} - 38 \, b^{4} c d^{2} x^{2} + 8 \, b^{5} d^{2} x\right )} e^{4} - {\left (12 \, b c^{4} d^{3} x^{4} + 46 \, b^{2} c^{3} d^{3} x^{3} + 11 \, b^{3} c^{2} d^{3} x^{2} - 23 \, b^{4} c d^{3} x\right )} e^{3} + {\left (6 \, c^{5} d^{4} x^{4} - 18 \, b c^{4} d^{4} x^{3} - 41 \, b^{2} c^{3} d^{4} x^{2} - 17 \, b^{3} c^{2} d^{4} x\right )} e^{2} + 12 \, {\left (c^{5} d^{5} x^{3} - b^{2} c^{3} d^{5} x\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) - 3 \, {\left (8 \, {\left (b^{3} c^{2} x^{4} + b^{4} c x^{3}\right )} e^{6} - {\left (19 \, b^{2} c^{3} d x^{4} + 3 \, b^{3} c^{2} d x^{3} - 16 \, b^{4} c d x^{2}\right )} e^{5} + {\left (9 \, b c^{4} d^{2} x^{4} - 29 \, b^{2} c^{3} d^{2} x^{3} - 30 \, b^{3} c^{2} d^{2} x^{2} + 8 \, b^{4} c d^{2} x\right )} e^{4} - {\left (6 \, c^{5} d^{3} x^{4} - 12 \, b c^{4} d^{3} x^{3} + b^{2} c^{3} d^{3} x^{2} + 19 \, b^{3} c^{2} d^{3} x\right )} e^{3} - 3 \, {\left (4 \, c^{5} d^{4} x^{3} + b c^{4} d^{4} x^{2} - 3 \, b^{2} c^{3} d^{4} x\right )} e^{2} - 6 \, {\left (c^{5} d^{5} x^{2} + b c^{4} d^{5} x\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) - 3 \, \sqrt {c x^{2} + b x} {\left (8 \, {\left (b^{3} c^{2} x^{3} + b^{4} c x^{2}\right )} e^{6} - {\left (19 \, b^{2} c^{3} d x^{3} + 7 \, b^{3} c^{2} d x^{2} - 12 \, b^{4} c d x\right )} e^{5} + {\left (9 \, b c^{4} d^{2} x^{3} - 20 \, b^{2} c^{3} d^{2} x^{2} - 26 \, b^{3} c^{2} d^{2} x + 3 \, b^{4} c d^{2}\right )} e^{4} - 3 \, {\left (2 \, c^{5} d^{3} x^{3} - 5 \, b c^{4} d^{3} x^{2} - 3 \, b^{2} c^{3} d^{3} x + 3 \, b^{3} c^{2} d^{3}\right )} e^{3} - 3 \, {\left (4 \, c^{5} d^{4} x^{2} - b c^{4} d^{4} x - 3 \, b^{2} c^{3} d^{4}\right )} e^{2} - 3 \, {\left (2 \, c^{5} d^{5} x + b c^{4} d^{5}\right )} e\right )} \sqrt {x e + d}\right )}}{9 \, {\left ({\left (b^{5} c^{2} d^{3} x^{4} + b^{6} c d^{3} x^{3}\right )} e^{6} - {\left (3 \, b^{4} c^{3} d^{4} x^{4} + b^{5} c^{2} d^{4} x^{3} - 2 \, b^{6} c d^{4} x^{2}\right )} e^{5} + {\left (3 \, b^{3} c^{4} d^{5} x^{4} - 3 \, b^{4} c^{3} d^{5} x^{3} - 5 \, b^{5} c^{2} d^{5} x^{2} + b^{6} c d^{5} x\right )} e^{4} - {\left (b^{2} c^{5} d^{6} x^{4} - 5 \, b^{3} c^{4} d^{6} x^{3} - 3 \, b^{4} c^{3} d^{6} x^{2} + 3 \, b^{5} c^{2} d^{6} x\right )} e^{3} - {\left (2 \, b^{2} c^{5} d^{7} x^{3} - b^{3} c^{4} d^{7} x^{2} - 3 \, b^{4} c^{3} d^{7} x\right )} e^{2} - {\left (b^{2} c^{5} d^{8} x^{2} + b^{3} c^{4} d^{8} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

2/9*((6*c^5*d^6*x^2 + 6*b*c^4*d^6*x - 8*(b^4*c*x^4 + b^5*x^3)*e^6 + (23*b^3*c^2*d*x^4 + 7*b^4*c*d*x^3 - 16*b^5
*d*x^2)*e^5 - (17*b^2*c^3*d^2*x^4 - 29*b^3*c^2*d^2*x^3 - 38*b^4*c*d^2*x^2 + 8*b^5*d^2*x)*e^4 - (12*b*c^4*d^3*x
^4 + 46*b^2*c^3*d^3*x^3 + 11*b^3*c^2*d^3*x^2 - 23*b^4*c*d^3*x)*e^3 + (6*c^5*d^4*x^4 - 18*b*c^4*d^4*x^3 - 41*b^
2*c^3*d^4*x^2 - 17*b^3*c^2*d^4*x)*e^2 + 12*(c^5*d^5*x^3 - b^2*c^3*d^5*x)*e)*sqrt(c)*e^(1/2)*weierstrassPInvers
e(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*
e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) - 3*(8*(b^3*c^2*x^4 + b^4*c*x^3)*e^6 - (19*b^2*c^3*d*x^4 + 3*b
^3*c^2*d*x^3 - 16*b^4*c*d*x^2)*e^5 + (9*b*c^4*d^2*x^4 - 29*b^2*c^3*d^2*x^3 - 30*b^3*c^2*d^2*x^2 + 8*b^4*c*d^2*
x)*e^4 - (6*c^5*d^3*x^4 - 12*b*c^4*d^3*x^3 + b^2*c^3*d^3*x^2 + 19*b^3*c^2*d^3*x)*e^3 - 3*(4*c^5*d^4*x^3 + b*c^
4*d^4*x^2 - 3*b^2*c^3*d^4*x)*e^2 - 6*(c^5*d^5*x^2 + b*c^4*d^5*x)*e)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d
^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3,
weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) - 3*sqrt(c*x^2 + b*x)*(8*(b^3*c^2*x^3 + b^4
*c*x^2)*e^6 - (19*b^2*c^3*d*x^3 + 7*b^3*c^2*d*x^2 - 12*b^4*c*d*x)*e^5 + (9*b*c^4*d^2*x^3 - 20*b^2*c^3*d^2*x^2
- 26*b^3*c^2*d^2*x + 3*b^4*c*d^2)*e^4 - 3*(2*c^5*d^3*x^3 - 5*b*c^4*d^3*x^2 - 3*b^2*c^3*d^3*x + 3*b^3*c^2*d^3)*
e^3 - 3*(4*c^5*d^4*x^2 - b*c^4*d^4*x - 3*b^2*c^3*d^4)*e^2 - 3*(2*c^5*d^5*x + b*c^4*d^5)*e)*sqrt(x*e + d))/((b^
5*c^2*d^3*x^4 + b^6*c*d^3*x^3)*e^6 - (3*b^4*c^3*d^4*x^4 + b^5*c^2*d^4*x^3 - 2*b^6*c*d^4*x^2)*e^5 + (3*b^3*c^4*
d^5*x^4 - 3*b^4*c^3*d^5*x^3 - 5*b^5*c^2*d^5*x^2 + b^6*c*d^5*x)*e^4 - (b^2*c^5*d^6*x^4 - 5*b^3*c^4*d^6*x^3 - 3*
b^4*c^3*d^6*x^2 + 3*b^5*c^2*d^6*x)*e^3 - (2*b^2*c^5*d^7*x^3 - b^3*c^4*d^7*x^2 - 3*b^4*c^3*d^7*x)*e^2 - (b^2*c^
5*d^8*x^2 + b^3*c^4*d^8*x)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(1/((x*(b + c*x))**(3/2)*(d + e*x)**(5/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(x*e + d)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^(3/2)*(d + e*x)^(5/2)),x)

[Out]

int(1/((b*x + c*x^2)^(3/2)*(d + e*x)^(5/2)), x)

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