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

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

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Rubi [A]  time = 0.47, 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 = {792, 672, 666, 660, 208} \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]

-(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]) - (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 208

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

Rule 660

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 666

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 672

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)), I
nt[(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] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 792

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)) \operatorname {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 [C]  time = 0.10, size = 129, normalized size = 0.42 \begin {gather*} \frac {\frac {c (d+e x)^2 (-4 b e g+3 c d g+5 c e f) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac {d g}{e}-f}{2 e (d+e x)^{3/2} (2 c d-b e) \sqrt {(d+e x) (c (d-e x)-b e)}} \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]

(-f + (d*g)/e + (c*(5*c*e*f + 3*c*d*g - 4*b*e*g)*(d + e*x)^2*Hypergeometric2F1[-1/2, 2, 1/2, (-(c*d) + b*e + c
*e*x)/(-2*c*d + b*e)])/(e*(-2*c*d + b*e)^2))/(2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[(d + e*x)*(-(b*e) + c*(d
- e*x))])

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IntegrateAlgebraic [A]  time = 6.52, size = 362, normalized size = 1.18 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (-4 b^2 e^2 g (d+e x)+2 b^2 d e^2 g-2 b^2 e^3 f-8 b c d^2 e g+5 b c e^2 f (d+e x)+8 b c d e^2 f+11 b c d e g (d+e x)-12 b c e g (d+e x)^2+8 c^2 d^3 g-8 c^2 d^2 e f-6 c^2 d^2 g (d+e x)-10 c^2 d e f (d+e x)+15 c^2 e f (d+e x)^2+9 c^2 d g (d+e x)^2\right )}{4 e^2 (d+e x)^{5/2} (b e-2 c d)^3 (b e+c (d+e x)-2 c d)}+\frac {3 \left (-4 b c e g+3 c^2 d g+5 c^2 e f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{4 e^2 (2 c d-b e)^3 \sqrt {b e-2 c d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]

(Sqrt[2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2]*(-8*c^2*d^2*e*f + 8*b*c*d*e^2*f - 2*b^2*e^3*f + 8*c^2*d
^3*g - 8*b*c*d^2*e*g + 2*b^2*d*e^2*g - 10*c^2*d*e*f*(d + e*x) + 5*b*c*e^2*f*(d + e*x) - 6*c^2*d^2*g*(d + e*x)
+ 11*b*c*d*e*g*(d + e*x) - 4*b^2*e^2*g*(d + e*x) + 15*c^2*e*f*(d + e*x)^2 + 9*c^2*d*g*(d + e*x)^2 - 12*b*c*e*g
*(d + e*x)^2))/(4*e^2*(-2*c*d + b*e)^3*(d + e*x)^(5/2)*(-2*c*d + b*e + c*(d + e*x))) + (3*(5*c^2*e*f + 3*c^2*d
*g - 4*b*c*e*g)*ArcTan[(Sqrt[-2*c*d + b*e]*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2])/(Sqrt[d + e*x]*(-2*c
*d + b*e + c*(d + e*x)))])/(4*e^2*(2*c*d - b*e)^3*Sqrt[-2*c*d + b*e])

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fricas [B]  time = 0.48, size = 1976, normalized size = 6.42

result too large to display

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*((5*c^3*e^5*f + (3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d*e^4 + b*c^2*e^5)*f + (6*c^3*d^2*e^3
- 5*b*c^2*d*e^4 - 4*b^2*c*e^5)*g)*x^3 + 3*(5*b*c^2*d*e^4*f + (3*b*c^2*d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3
*d^4*e - b*c^2*d^3*e^2)*f - (3*c^3*d^5 - 7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g - (5*(2*c^3*d^3*e^2 - 3*b*c^2*d^2*
e^3)*f + (6*c^3*d^4*e - 17*b*c^2*d^3*e^2 + 12*b^2*c*d^2*e^3)*g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2
+ 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d
))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(5*(2*c^3*d*e^3 - b*c^2*e^4)*f
+ (6*c^3*d^2*e^2 - 11*b*c^2*d*e^3 + 4*b^2*c*e^4)*g)*x^2 - (6*c^3*d^3*e - 29*b*c^2*d^2*e^2 + 17*b^2*c*d*e^3 -
2*b^3*e^4)*f + (22*c^3*d^4 - 29*b*c^2*d^3*e + 5*b^2*c*d^2*e^2 + 2*b^3*d*e^3)*g + (5*(8*c^3*d^2*e^2 - 2*b*c^2*d
*e^3 - b^2*c*e^4)*f + (24*c^3*d^3*e - 38*b*c^2*d^2*e^2 + 5*b^2*c*d*e^3 + 4*b^3*e^4)*g)*x)*sqrt(e*x + d))/(16*c
^5*d^8*e^2 - 48*b*c^4*d^7*e^3 + 56*b^2*c^3*d^6*e^4 - 32*b^3*c^2*d^5*e^5 + 9*b^4*c*d^4*e^6 - b^5*d^3*e^7 - (16*
c^5*d^4*e^6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^9 + b^4*c*e^10)*x^4 - (32*c^5*d^5*e^5 - 48
*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2*d^2*e^8 - 6*b^4*c*d*e^9 + b^5*e^10)*x^3 - 3*(16*b*c^4*d^5*e^5
- 32*b^2*c^3*d^4*e^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*e^9)*x^2 + (32*c^5*d^7*e^3 - 112*b*c^4*d^6
*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6 + 26*b^4*c*d^3*e^7 - 3*b^5*d^2*e^8)*x), 1/4*(3*((5*c^3*e^5*f +
(3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d*e^4 + b*c^2*e^5)*f + (6*c^3*d^2*e^3 - 5*b*c^2*d*e^4 - 4*b^2*
c*e^5)*g)*x^3 + 3*(5*b*c^2*d*e^4*f + (3*b*c^2*d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3*d^4*e - b*c^2*d^3*e^2)*
f - (3*c^3*d^5 - 7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g - (5*(2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3)*f + (6*c^3*d^4*e -
17*b*c^2*d^3*e^2 + 12*b^2*c*d^2*e^3)*g)*x)*sqrt(-2*c*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e
)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2
- b*d*e)*(3*(5*(2*c^3*d*e^3 - b*c^2*e^4)*f + (6*c^3*d^2*e^2 - 11*b*c^2*d*e^3 + 4*b^2*c*e^4)*g)*x^2 - (6*c^3*d^
3*e - 29*b*c^2*d^2*e^2 + 17*b^2*c*d*e^3 - 2*b^3*e^4)*f + (22*c^3*d^4 - 29*b*c^2*d^3*e + 5*b^2*c*d^2*e^2 + 2*b^
3*d*e^3)*g + (5*(8*c^3*d^2*e^2 - 2*b*c^2*d*e^3 - b^2*c*e^4)*f + (24*c^3*d^3*e - 38*b*c^2*d^2*e^2 + 5*b^2*c*d*e
^3 + 4*b^3*e^4)*g)*x)*sqrt(e*x + d))/(16*c^5*d^8*e^2 - 48*b*c^4*d^7*e^3 + 56*b^2*c^3*d^6*e^4 - 32*b^3*c^2*d^5*
e^5 + 9*b^4*c*d^4*e^6 - b^5*d^3*e^7 - (16*c^5*d^4*e^6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^
9 + b^4*c*e^10)*x^4 - (32*c^5*d^5*e^5 - 48*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2*d^2*e^8 - 6*b^4*c*d*
e^9 + b^5*e^10)*x^3 - 3*(16*b*c^4*d^5*e^5 - 32*b^2*c^3*d^4*e^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*
e^9)*x^2 + (32*c^5*d^7*e^3 - 112*b*c^4*d^6*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6 + 26*b^4*c*d^3*e^7 -
3*b^5*d^2*e^8)*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.47Unable to transpose Error: Bad Argument Value

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maple [B]  time = 0.09, size = 824, normalized size = 2.68 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (12 \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-9 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-15 \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+24 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+12 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}-18 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-30 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-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}+4 \sqrt {b e -2 c d}\, b^{2} e^{3} g x +12 \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+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 -9 \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-15 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-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 +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 ) \left (b e -2 c d \right )^{\frac {7}{2}} e^{2}} \end {gather*}

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)

[Out]

-1/4/(e*x+d)^(5/2)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(12*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(
1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c*e^3*g-9*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*x^2*c^2*d*e^2*g-15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^2*e^3*f+24*
(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c*d*e^2*g-18*(-c*e*x-b*e+c*d)^(1/2
)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*g-30*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f+12*(b*e-2*c*d)^(1/2)*x^2*b*c*e^3*g-9*(b*e-2*c*d)^(1/2)*x^2*c^2*d*e
^2*g-15*(b*e-2*c*d)^(1/2)*x^2*c^2*e^3*f+12*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*b*c*d^2*e*g-9*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-15*(-c*e*
x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*(b*e-2*c*d)^(1/2)*x*b^2*e^3*g+
13*(b*e-2*c*d)^(1/2)*x*b*c*d*e^2*g-5*(b*e-2*c*d)^(1/2)*x*b*c*e^3*f-12*(b*e-2*c*d)^(1/2)*x*c^2*d^2*e*g-20*(b*e-
2*c*d)^(1/2)*x*c^2*d*e^2*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*} \int \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(3/2)), x)

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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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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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