3.23.66 \(\int \frac {f+g x}{(d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [2266]

Optimal. Leaf size=153 \[ -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(c e f+3 c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}} \]

[Out]

-(-2*b*e*g+3*c*d*g+c*e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2))/e^2
/(-b*e+2*c*d)^(3/2)-(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)/(e*x+d)^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {806, 674, 214} \begin {gather*} -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac {(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^(3/2)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

-(((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2))) - ((c*e*f + 3*c
*d*g - 2*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x])])/(e^2*(2*
c*d - b*e)^(3/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {f+g x}{(d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(c e f+3 c d g-2 b e g) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(c e f+3 c d g-2 b e g) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )}{2 c d-b e}\\ &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(c e f+3 c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.40, size = 153, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {(e f-d g) (-c d+b e+c e x)}{(2 c d-b e) (d+e x)}-\frac {(c e f+3 c d g-2 b e g) \sqrt {-b e+c (d-e x)} \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2}}\right )}{e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^(3/2)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(Sqrt[d + e*x]*(((e*f - d*g)*(-(c*d) + b*e + c*e*x))/((2*c*d - b*e)*(d + e*x)) - ((c*e*f + 3*c*d*g - 2*b*e*g)*
Sqrt[-(b*e) + c*(d - e*x)]*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^(3/2)))/(e^2*
Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(319\) vs. \(2(141)=282\).
time = 0.04, size = 320, normalized size = 2.09

method	result	size
default	\(\frac {\left (2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,e^{2} g x -3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e g x -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,e^{2} f x +2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b d e g -3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,d^{2} g -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e f -\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, d g +\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, e f \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{\left (b e -2 c d \right )^{\frac {3}{2}} e^{2} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {3}{2}}}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*e^2*g*x-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2
))*c*d*e*g*x-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*e^2*f*x+2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2
*c*d)^(1/2))*b*d*e*g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*d^2*g-arctan((-c*e*x-b*e+c*d)^(1/2)/
(b*e-2*c*d)^(1/2))*c*d*e*f-(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*d*g+(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/
2)*e*f)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(3/2)

________________________________________________________________________________________

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((g*x+f)/(e*x+d)^(3/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)/(sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(x*e + d)^(3/2)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (144) = 288\).
time = 3.34, size = 642, normalized size = 4.20 \begin {gather*} \left [\frac {{\left (3 \, c d^{3} g + {\left (c f - 2 \, b g\right )} x^{2} e^{3} + {\left (3 \, c d g x^{2} + 2 \, {\left (c d f - 2 \, b d g\right )} x\right )} e^{2} + {\left (6 \, c d^{2} g x + c d^{2} f - 2 \, b d^{2} g\right )} e\right )} \sqrt {2 \, c d - b e} \log \left (\frac {3 \, c d^{2} - {\left (c x^{2} + 2 \, b x\right )} e^{2} + 2 \, {\left (c d x - b d\right )} e - 2 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {2 \, c d - b e} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (2 \, c d^{2} g + b f e^{2} - {\left (2 \, c d f + b d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{2 \, {\left (4 \, c^{2} d^{4} e^{2} + b^{2} x^{2} e^{6} - 2 \, {\left (2 \, b c d x^{2} - b^{2} d x\right )} e^{5} + {\left (4 \, c^{2} d^{2} x^{2} - 8 \, b c d^{2} x + b^{2} d^{2}\right )} e^{4} + 4 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\right )} e^{3}\right )}}, -\frac {{\left (3 \, c d^{3} g + {\left (c f - 2 \, b g\right )} x^{2} e^{3} + {\left (3 \, c d g x^{2} + 2 \, {\left (c d f - 2 \, b d g\right )} x\right )} e^{2} + {\left (6 \, c d^{2} g x + c d^{2} f - 2 \, b d^{2} g\right )} e\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-2 \, c d + b e} \sqrt {x e + d}}{\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}\right ) - {\left (2 \, c d^{2} g + b f e^{2} - {\left (2 \, c d f + b d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{4 \, c^{2} d^{4} e^{2} + b^{2} x^{2} e^{6} - 2 \, {\left (2 \, b c d x^{2} - b^{2} d x\right )} e^{5} + {\left (4 \, c^{2} d^{2} x^{2} - 8 \, b c d^{2} x + b^{2} d^{2}\right )} e^{4} + 4 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\right )} e^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((3*c*d^3*g + (c*f - 2*b*g)*x^2*e^3 + (3*c*d*g*x^2 + 2*(c*d*f - 2*b*d*g)*x)*e^2 + (6*c*d^2*g*x + c*d^2*f
- 2*b*d^2*g)*e)*sqrt(2*c*d - b*e)*log((3*c*d^2 - (c*x^2 + 2*b*x)*e^2 + 2*(c*d*x - b*d)*e - 2*sqrt(c*d^2 - b*d*
e - (c*x^2 + b*x)*e^2)*sqrt(2*c*d - b*e)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(2*c*d^2*g + b*f*e^2 -
(2*c*d*f + b*d*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(4*c^2*d^4*e^2 + b^2*x^2*e^6 - 2*(
2*b*c*d*x^2 - b^2*d*x)*e^5 + (4*c^2*d^2*x^2 - 8*b*c*d^2*x + b^2*d^2)*e^4 + 4*(2*c^2*d^3*x - b*c*d^3)*e^3), -((
3*c*d^3*g + (c*f - 2*b*g)*x^2*e^3 + (3*c*d*g*x^2 + 2*(c*d*f - 2*b*d*g)*x)*e^2 + (6*c*d^2*g*x + c*d^2*f - 2*b*d
^2*g)*e)*sqrt(-2*c*d + b*e)*arctan(-sqrt(-2*c*d + b*e)*sqrt(x*e + d)/sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))
- (2*c*d^2*g + b*f*e^2 - (2*c*d*f + b*d*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(4*c^2*d^
4*e^2 + b^2*x^2*e^6 - 2*(2*b*c*d*x^2 - b^2*d*x)*e^5 + (4*c^2*d^2*x^2 - 8*b*c*d^2*x + b^2*d^2)*e^4 + 4*(2*c^2*d
^3*x - b*c*d^3)*e^3)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**(3/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**(3/2)), x)

________________________________________________________________________________________

Giac [A]
time = 1.69, size = 168, normalized size = 1.10 \begin {gather*} \frac {{\left (\frac {{\left (3 \, c^{2} d g + c^{2} f e - 2 \, b c g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} f e}{{\left (2 \, c d - b e\right )} {\left (x e + d\right )} c}\right )} e^{\left (-2\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*c^2*d*g + c^2*f*e - 2*b*c*g*e)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((2*c*d - b*e)*
sqrt(-2*c*d + b*e)) + (sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^2*d*g - sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^2*f*e)/((
2*c*d - b*e)*(x*e + d)*c))*e^(-2)/c

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f+g\,x}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________