3.23.74 \(\int \frac {f+g x}{(d+e x)^{3/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [2274]

Optimal. Leaf size=308 \[ \frac {-e f+d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c (5 c e f+3 c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {3 c (5 c e f+3 c d g-4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {806, 686, 680, 674, 214} \begin {gather*} -\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c \sqrt {d+e x} (-4 b e g+3 c d g+5 c e f)}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {-4 b e g+3 c d g+5 c e f}{4 e^2 \sqrt {d+e x} (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {3 c (-4 b e g+3 c d g+5 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 )}{4 e^2 (2 c d-b e)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c*e*f + 3*
c*d*g - 4*b*e*g)/(4*e^2*(2*c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (3*c*(5*c*e
*f + 3*c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(4*e^2*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (3*
c*(5*c*e*f + 3*c*d*g - 4*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])])/(4*e^2*(2*c*d - b*e)^(7/2))

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 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))),
Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

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} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {e f-d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f+3 c d g-4 b e g) \int \frac {1}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(3 c (5 c e f+3 c d g-4 b e g)) \int \frac {\sqrt {d+e x}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{8 e (2 c d-b e)^2}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c (5 c e f+3 c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(3 c (5 c e f+3 c d g-4 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}{8 e (2 c d-b e)^3}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c (5 c e f+3 c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(3 c (5 c e f+3 c d g-4 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 )}{4 (2 c d-b e)^3}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c (5 c e f+3 c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {3 c (5 c e f+3 c d g-4 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 )}{4 e^2 (2 c d-b e)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.97, size = 255, normalized size = 0.83 \begin {gather*} \frac {c (d+e x)^{3/2} \left (\frac {(-b e+c (d-e x)) \left (b c e \left (-9 d^2 g+e^2 x (5 f-12 g x)+13 d e (f-g x)\right )-2 b^2 e^2 (d g+e (f+2 g x))+c^2 \left (11 d^3 g+15 e^3 f x^2-3 d^2 e (f-4 g x)+d e^2 x (20 f+9 g x)\right )\right )}{c (2 c d-b e)^3 (d+e x)^2}-\frac {3 (5 c e f+3 c d g-4 b e g) (-b e+c (d-e x))^{3/2} \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2}}\right )}{4 e^2 ((d+e x) (-b e+c (d-e x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^(3/2)*(((-(b*e) + c*(d - e*x))*(b*c*e*(-9*d^2*g + e^2*x*(5*f - 12*g*x) + 13*d*e*(f - g*x)) - 2*b^
2*e^2*(d*g + e*(f + 2*g*x)) + c^2*(11*d^3*g + 15*e^3*f*x^2 - 3*d^2*e*(f - 4*g*x) + d*e^2*x*(20*f + 9*g*x))))/(
c*(2*c*d - b*e)^3*(d + e*x)^2) - (3*(5*c*e*f + 3*c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x))^(3/2)*ArcTan[Sqrt[-(b
*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^(7/2)))/(4*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(815\) vs. \(2(280)=560\).
time = 0.05, size = 816, normalized size = 2.65

method	result	size
default	\(-\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}-9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+12 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}-9 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+24 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x -30 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x +4 \sqrt {b e -2 c d}\, b^{2} e^{3} g x +13 \sqrt {b e -2 c d}\, b c d \,e^{2} g x -5 \sqrt {b e -2 c d}\, b c \,e^{3} f x -12 \sqrt {b e -2 c d}\, c^{2} d^{2} e g x -20 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +12 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f +2 \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +2 \sqrt {b e -2 c d}\, b^{2} e^{3} f +9 \sqrt {b e -2 c d}\, b c \,d^{2} e g -13 \sqrt {b e -2 c d}\, b c d \,e^{2} f -11 \sqrt {b e -2 c d}\, c^{2} d^{3} g +3 \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (c e x +b e -c d \right ) e^{2} \left (b e -2 c d \right )^{\frac {7}{2}}}\)	\(816\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/(e*x+d)^(5/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(12*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e
*x-b*e+c*d)^(1/2)*b*c*e^3*g*x^2-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*
d*e^2*g*x^2-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*e^3*f*x^2+12*(b*e-2
*c*d)^(1/2)*b*c*e^3*g*x^2-9*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2-15*(b*e-2*c*d)^(1/2)*c^2*e^3*f*x^2+24*arctan((-c
*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*c*d*e^2*g*x-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(
b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d^2*e*g*x-30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c
*e*x-b*e+c*d)^(1/2)*c^2*d*e^2*f*x+4*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x+13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*g*x-5*(b*e-2*
c*d)^(1/2)*b*c*e^3*f*x-12*(b*e-2*c*d)^(1/2)*c^2*d^2*e*g*x-20*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x+12*arctan((-c*e*x
-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*c*d^2*e*g-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c
*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d^3*g-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*
d)^(1/2)*c^2*d^2*e*f+2*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+2*(b*e-2*c*d)^(1/2)*b^2*e^3*f+9*(b*e-2*c*d)^(1/2)*b*c*d^2
*e*g-13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f-11*(b*e-2*c*d)^(1/2)*c^2*d^3*g+3*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f)/(c*e*x+b
*e-c*d)/e^2/(b*e-2*c*d)^(7/2)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (288) = 576\).
time = 2.85, size = 1862, normalized size = 6.05 \begin {gather*} \text {Too large to display} \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)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(3*(3*c^3*d^5*g - ((5*c^3*f - 4*b*c^2*g)*x^4 + (5*b*c^2*f - 4*b^2*c*g)*x^3)*e^5 - (3*c^3*d*g*x^4 + 5*(2*c
^3*d*f - b*c^2*d*g)*x^3 + 3*(5*b*c^2*d*f - 4*b^2*c*d*g)*x^2)*e^4 - 3*(2*c^3*d^2*g*x^3 + 3*b*c^2*d^2*g*x^2 + (5
*b*c^2*d^2*f - 4*b^2*c*d^2*g)*x)*e^3 - (5*b*c^2*d^3*f - 4*b^2*c*d^3*g - (10*c^3*d^3*f - 17*b*c^2*d^3*g)*x)*e^2
+ (6*c^3*d^4*g*x + 5*c^3*d^4*f - 7*b*c^2*d^4*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*(22*c^3*d^4*g + (2*b^3*f - 3*(5*b*c^2*f - 4*b^2*c*g)*x^2 - (5*b^2*c*f - 4*b^3*g)*x)*e^4 - (17*b^2
*c*d*f - 2*b^3*d*g - 3*(10*c^3*d*f - 11*b*c^2*d*g)*x^2 + 5*(2*b*c^2*d*f - b^2*c*d*g)*x)*e^3 + (18*c^3*d^2*g*x^
2 + 29*b*c^2*d^2*f + 5*b^2*c*d^2*g + 2*(20*c^3*d^2*f - 19*b*c^2*d^2*g)*x)*e^2 + (24*c^3*d^3*g*x - 6*c^3*d^3*f
- 29*b*c^2*d^3*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(16*c^5*d^8*e^2 - (b^4*c*x^4 + b^5
*x^3)*e^10 + (8*b^3*c^2*d*x^4 + 6*b^4*c*d*x^3 - 3*b^5*d*x^2)*e^9 - (24*b^2*c^3*d^2*x^4 + 8*b^3*c^2*d^2*x^3 - 2
4*b^4*c*d^2*x^2 + 3*b^5*d^2*x)*e^8 + (32*b*c^4*d^3*x^4 - 16*b^2*c^3*d^3*x^3 - 72*b^3*c^2*d^3*x^2 + 26*b^4*c*d^
3*x - b^5*d^3)*e^7 - (16*c^5*d^4*x^4 - 48*b*c^4*d^4*x^3 - 96*b^2*c^3*d^4*x^2 + 88*b^3*c^2*d^4*x - 9*b^4*c*d^4)
*e^6 - 16*(2*c^5*d^5*x^3 + 3*b*c^4*d^5*x^2 - 9*b^2*c^3*d^5*x + 2*b^3*c^2*d^5)*e^5 - 56*(2*b*c^4*d^6*x - b^2*c^
3*d^6)*e^4 + 16*(2*c^5*d^7*x - 3*b*c^4*d^7)*e^3), -1/4*(3*(3*c^3*d^5*g - ((5*c^3*f - 4*b*c^2*g)*x^4 + (5*b*c^2
*f - 4*b^2*c*g)*x^3)*e^5 - (3*c^3*d*g*x^4 + 5*(2*c^3*d*f - b*c^2*d*g)*x^3 + 3*(5*b*c^2*d*f - 4*b^2*c*d*g)*x^2)
*e^4 - 3*(2*c^3*d^2*g*x^3 + 3*b*c^2*d^2*g*x^2 + (5*b*c^2*d^2*f - 4*b^2*c*d^2*g)*x)*e^3 - (5*b*c^2*d^3*f - 4*b^
2*c*d^3*g - (10*c^3*d^3*f - 17*b*c^2*d^3*g)*x)*e^2 + (6*c^3*d^4*g*x + 5*c^3*d^4*f - 7*b*c^2*d^4*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)) - (22*c^3*d^4*g +
(2*b^3*f - 3*(5*b*c^2*f - 4*b^2*c*g)*x^2 - (5*b^2*c*f - 4*b^3*g)*x)*e^4 - (17*b^2*c*d*f - 2*b^3*d*g - 3*(10*c
^3*d*f - 11*b*c^2*d*g)*x^2 + 5*(2*b*c^2*d*f - b^2*c*d*g)*x)*e^3 + (18*c^3*d^2*g*x^2 + 29*b*c^2*d^2*f + 5*b^2*c
*d^2*g + 2*(20*c^3*d^2*f - 19*b*c^2*d^2*g)*x)*e^2 + (24*c^3*d^3*g*x - 6*c^3*d^3*f - 29*b*c^2*d^3*g)*e)*sqrt(c*
d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(16*c^5*d^8*e^2 - (b^4*c*x^4 + b^5*x^3)*e^10 + (8*b^3*c^2*d*x^
4 + 6*b^4*c*d*x^3 - 3*b^5*d*x^2)*e^9 - (24*b^2*c^3*d^2*x^4 + 8*b^3*c^2*d^2*x^3 - 24*b^4*c*d^2*x^2 + 3*b^5*d^2*
x)*e^8 + (32*b*c^4*d^3*x^4 - 16*b^2*c^3*d^3*x^3 - 72*b^3*c^2*d^3*x^2 + 26*b^4*c*d^3*x - b^5*d^3)*e^7 - (16*c^5
*d^4*x^4 - 48*b*c^4*d^4*x^3 - 96*b^2*c^3*d^4*x^2 + 88*b^3*c^2*d^4*x - 9*b^4*c*d^4)*e^6 - 16*(2*c^5*d^5*x^3 + 3
*b*c^4*d^5*x^2 - 9*b^2*c^3*d^5*x + 2*b^3*c^2*d^5)*e^5 - 56*(2*b*c^4*d^6*x - b^2*c^3*d^6)*e^4 + 16*(2*c^5*d^7*x
- 3*b*c^4*d^7)*e^3)]

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

[Out]

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

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Giac [A]
time = 2.86, size = 484, normalized size = 1.57 \begin {gather*} \frac {3 \, {\left (3 \, c^{2} d g + 5 \, c^{2} f e - 4 \, 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 )}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, {\left (c^{2} d g + c^{2} f e - b c g e\right )}}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} + \frac {2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 18 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e + 7 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 9 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2} - 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c g e^{2} + 7 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} f e - 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c g e}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left (x e + d\right )}^{2} c^{2}} \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)^(3/2),x, algorithm="giac")

[Out]

3/4*(3*c^2*d*g + 5*c^2*f*e - 4*b*c*g*e)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((8*c^3*d^
3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*sqrt(-2*c*d + b*e)) + 2*(c^2*d*g + c^2*f*e - b*c*g*e)/((8*
c^3*d^3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*sqrt(-(x*e + d)*c + 2*c*d - b*e)) + 1/4*(2*sqrt(-(x*
e + d)*c + 2*c*d - b*e)*c^3*d^2*g - 18*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^3*d*f*e + 7*sqrt(-(x*e + d)*c + 2*c*
d - b*e)*b*c^2*d*g*e + (-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 9*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c^2*f
*e^2 - 4*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b^2*c*g*e^2 + 7*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*f*e - 4*(-(x*
e + d)*c + 2*c*d - b*e)^(3/2)*b*c*g*e)/((8*c^3*d^3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*(x*e + d)
^2*c^2)

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

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