3.5.26 \(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)^{5/2}} \, dx\) [426]

Optimal. Leaf size=451 \[ -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {754, 836, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 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)^{7/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {4 \sqrt {c} \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 (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d +
e*x]*(b*(c*d - b*e)*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + 2*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)
*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (4*Sqrt[c]*(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])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7
/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sq
rt[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)*d*(c*d - b*e)*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 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -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}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+\frac {3}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b c d e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-\frac {1}{2} c 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 d^2 (c d-b e)^2}\\ &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \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 d (c d-b e)}-\frac {\left (2 c (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 d^2 (c d-b e)^2}\\ &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \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 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (2 c (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 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (2 c (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 d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \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 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} (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} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \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} d (c d-b e) \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 = 8.54, size = 429, normalized size = 0.95 \begin {gather*} \frac {2 \left (b (d+e x) \left (b c^3 d^2 (c d-b e) x^2+2 c^3 d^2 (4 c d-5 b e) x^2 (b+c x)-b d (c d-b e)^2 (b+c x)^2+2 (c d-b e)^2 (4 c d+b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} 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 \sinh ^{-1}\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} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^2 (c d-b e)^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1762\) vs. \(2(397)=794\).
time = 0.49, size = 1763, normalized size = 3.91

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 b^{3} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )^{2}}-\frac {4 \left (c e \,x^{2}+c d x \right ) c^{2} \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 d \,b^{3} x^{2}}+\frac {4 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e +4 c d \right )}{3 b^{4} d^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (-\frac {c^{2} e}{3 \left (b e -c d \right ) b^{3}}+\frac {2 c^{2} \left (5 b e -4 c d \right )}{3 \left (b e -c d \right ) b^{4}}+\frac {2 c^{3} d \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2}}-\frac {c e}{3 b^{3} d}\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 {2 c^{3} e \left (5 b e -4 c d \right )}{3 \left (b e -c d \right )^{2} b^{4}}-\frac {2 c e \left (b e +4 c d \right )}{3 b^{4} 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 ) \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}}\)	\(653\)
default	\(\text {Expression too large to display}\)	\(1763\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(-b^3*c^3*d^4+2*x^4*b^3*c^3*e^4+16*x^4*c^6*d^3*e+4*x^3*b^4*c^2*e^4+24*x^2*b*c^5*d^4+6*x*b^2*c^4*d^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))*x^2
*b^4*c^2*d*e^3+15*((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))*x^2*b^3*c^3*d^2*e^2-32*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^4*d^3*e+2*((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))*x^2*b^4*c^2*d*e^3-28*((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))*x^2*b^3*c^3*d^2*
e^2+40*((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))*x^2*b^2*c^4*d^3*e+((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))*x*b^5*c*d*e^3+15*((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))*x*b^4*c^2*d^2*e^2-32*((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))*x*b^3*c^3*d^3*e+2*((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))*x*b^5*c*d*e^3-
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))*x*b^4*c^2*d^2*e^2+40*((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))*x*b^3*c^3*d^3*e-16*((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))*x*b^2*c^4*d^4+16*((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))*x^2*b*c^5*d^4+2*((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))*x^2*b^5*c*e^4-16*((
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))
*x^2*b*c^5*d^4+16*((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))*x*b^2*c^4*d^4+4*x^4*b^2*c^4*d*e^3-11*x*b^3*c^3*d^3*e+16*x^3*c^6*d^4+2*x^2*b^5*c*e^4+x*b^5
*c*d*e^3+2*((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))*x*b^6*e^4+6*x^2*b^4*c^2*d*e^3-3*x^2*b^3*c^3*d^2*e^2-31*x^2*b^2*c^4*d^3*e-24*x^4*b*c^5*d^2*e^2+9*
x^3*b^3*c^3*d*e^3-33*x^3*b^2*c^4*d^2*e^2+2*b^4*c^2*d^3*e+4*x*b^4*c^2*d^2*e^2-b^5*c*d^2*e^2)/x^2*(x*(c*x+b))^(1
/2)/d^2/c/b^4/(c*x+b)^2/(b*e-c*d)^2/(e*x+d)^(1/2)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.60, size = 959, normalized size = 2.13 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{6} d^{4} x^{4} + 32 \, b c^{5} d^{4} x^{3} + 16 \, b^{2} c^{4} d^{4} x^{2} + 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} e^{4} + 3 \, {\left (b^{3} c^{3} d x^{4} + 2 \, b^{4} c^{2} d x^{3} + b^{5} c d x^{2}\right )} e^{3} + 13 \, {\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{3} c^{3} d^{2} x^{3} + b^{4} c^{2} d^{2} x^{2}\right )} e^{2} - 32 \, {\left (b c^{5} d^{3} x^{4} + 2 \, b^{2} c^{4} d^{3} x^{3} + b^{3} c^{3} d^{3} x^{2}\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 ) + 6 \, {\left ({\left (b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{3} + b^{5} c x^{2}\right )} e^{4} + 2 \, {\left (b^{2} c^{4} d x^{4} + 2 \, b^{3} c^{3} d x^{3} + b^{4} c^{2} d x^{2}\right )} e^{3} - 12 \, {\left (b c^{5} d^{2} x^{4} + 2 \, b^{2} c^{4} d^{2} x^{3} + b^{3} c^{3} d^{2} x^{2}\right )} e^{2} + 8 \, {\left (c^{6} d^{3} x^{4} + 2 \, b c^{5} d^{3} x^{3} + b^{2} c^{4} d^{3} x^{2}\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 (2 \, {\left (b^{3} c^{3} x^{3} + 2 \, b^{4} c^{2} x^{2} + b^{5} c x\right )} e^{4} + {\left (4 \, b^{2} c^{4} d x^{3} + 7 \, b^{3} c^{3} d x^{2} + 2 \, b^{4} c^{2} d x - b^{5} c d\right )} e^{3} - {\left (24 \, b c^{5} d^{2} x^{3} + 37 \, b^{2} c^{4} d^{2} x^{2} + 10 \, b^{3} c^{3} d^{2} x - 2 \, b^{4} c^{2} d^{2}\right )} e^{2} + {\left (16 \, c^{6} d^{3} x^{3} + 24 \, b c^{5} d^{3} x^{2} + 6 \, b^{2} c^{4} d^{3} x - b^{3} c^{3} d^{3}\right )} e\right )} \sqrt {x e + d}\right )}}{9 \, {\left ({\left (b^{6} c^{3} d^{2} x^{4} + 2 \, b^{7} c^{2} d^{2} x^{3} + b^{8} c d^{2} x^{2}\right )} e^{3} - 2 \, {\left (b^{5} c^{4} d^{3} x^{4} + 2 \, b^{6} c^{3} d^{3} x^{3} + b^{7} c^{2} d^{3} x^{2}\right )} e^{2} + {\left (b^{4} c^{5} d^{4} x^{4} + 2 \, b^{5} c^{4} d^{4} x^{3} + b^{6} c^{3} d^{4} x^{2}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*((16*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 + 16*b^2*c^4*d^4*x^2 + 2*(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^4 + 3
*(b^3*c^3*d*x^4 + 2*b^4*c^2*d*x^3 + b^5*c*d*x^2)*e^3 + 13*(b^2*c^4*d^2*x^4 + 2*b^3*c^3*d^2*x^3 + b^4*c^2*d^2*x
^2)*e^2 - 32*(b*c^5*d^3*x^4 + 2*b^2*c^4*d^3*x^3 + b^3*c^3*d^3*x^2)*e)*sqrt(c)*e^(1/2)*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) + 6*((b^3*c^3*x^4 + 2*b^4*c^2*x^3 + b^5*c*x^2)*e^4 + 2*(b^2*c^4*d*x^
4 + 2*b^3*c^3*d*x^3 + b^4*c^2*d*x^2)*e^3 - 12*(b*c^5*d^2*x^4 + 2*b^2*c^4*d^2*x^3 + b^3*c^3*d^2*x^2)*e^2 + 8*(c
^6*d^3*x^4 + 2*b*c^5*d^3*x^3 + b^2*c^4*d^3*x^2)*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, weierstrassPInver
se(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)*(2*(b^3*c^3*x^3 + 2*b^4*c^2*x^2 + b^5*
c*x)*e^4 + (4*b^2*c^4*d*x^3 + 7*b^3*c^3*d*x^2 + 2*b^4*c^2*d*x - b^5*c*d)*e^3 - (24*b*c^5*d^2*x^3 + 37*b^2*c^4*
d^2*x^2 + 10*b^3*c^3*d^2*x - 2*b^4*c^2*d^2)*e^2 + (16*c^6*d^3*x^3 + 24*b*c^5*d^3*x^2 + 6*b^2*c^4*d^3*x - b^3*c
^3*d^3)*e)*sqrt(x*e + d))/((b^6*c^3*d^2*x^4 + 2*b^7*c^2*d^2*x^3 + b^8*c*d^2*x^2)*e^3 - 2*(b^5*c^4*d^3*x^4 + 2*
b^6*c^3*d^3*x^3 + b^7*c^2*d^3*x^2)*e^2 + (b^4*c^5*d^4*x^4 + 2*b^5*c^4*d^4*x^3 + b^6*c^3*d^4*x^2)*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 {5}{2}} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((c*x^2 + b*x)^(5/2)*sqrt(x*e + d)), 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 )}^{5/2}\,\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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