3.21.8 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=307 \[ \frac {c^2 (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{5/2}}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)} \]

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Rubi [A]  time = 0.53, antiderivative size = 307, 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, 662, 672, 660, 208} \begin {gather*} \frac {c^2 (-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 )}{8 e^2 (2 c d-b e)^{5/2}}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2),x]

[Out]

-((c*e*f + 3*c*d*g - 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*(2*c*d - b*e)*(d + e*x)^(5/2))
+ (c*(c*e*f + 3*c*d*g - 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e^2*(2*c*d - b*e)^2*(d + e*x)^
(3/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) + (c^
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])])/(8*e^2*(2*c*d - b*e)^(5/2))

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 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(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] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && 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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(c e f+3 c d g-2 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(c (c e f+3 c d g-2 b e g)) \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e (2 c d-b e)}\\ &=-\frac {(c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {c (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {\left (c^2 (c e f+3 c d g-2 b e g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e (2 c d-b e)^2}\\ &=-\frac {(c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {c (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {\left (c^2 (c e f+3 c d g-2 b e g)\right ) \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 )}{8 (2 c d-b e)^2}\\ &=-\frac {(c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {c (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {c^2 (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 )}{8 e^2 (2 c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.13, size = 128, normalized size = 0.42 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{3/2} \left (-\frac {3 c^2 (d+e x)^3 (-2 b e g+3 c d g+c e f) \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{(2 c d-b e)^3}+3 d g-3 e f\right )}{9 e^2 (d+e x)^{9/2} (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-3*e*f + 3*d*g - (3*c^2*(c*e*f + 3*c*d*g - 2*b*e*g)*(d + e*x)^3*Hyp
ergeometric2F1[3/2, 3, 5/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(2*c*d - b*e)^3))/(9*e^2*(2*c*d - b*e)*(d
+ e*x)^(9/2))

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IntegrateAlgebraic [A]  time = 1.89, size = 347, normalized size = 1.13 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (-12 b^2 e^2 g (d+e x)+8 b^2 d e^2 g-8 b^2 e^3 f-32 b c d^2 e g-2 b c e^2 f (d+e x)+32 b c d e^2 f+50 b c d e g (d+e x)-6 b c e g (d+e x)^2+32 c^2 d^3 g-32 c^2 d^2 e f-52 c^2 d^2 g (d+e x)+4 c^2 d e f (d+e x)+3 c^2 e f (d+e x)^2+9 c^2 d g (d+e x)^2\right )}{24 e^2 (d+e x)^{7/2} (b e-2 c d)^2}+\frac {\left (2 b c^2 e g-3 c^3 d g+c^3 (-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 )}{8 e^2 (2 c d-b e)^2 \sqrt {b e-2 c d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2),x]

[Out]

(Sqrt[2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2]*(-32*c^2*d^2*e*f + 32*b*c*d*e^2*f - 8*b^2*e^3*f + 32*c^
2*d^3*g - 32*b*c*d^2*e*g + 8*b^2*d*e^2*g + 4*c^2*d*e*f*(d + e*x) - 2*b*c*e^2*f*(d + e*x) - 52*c^2*d^2*g*(d + e
*x) + 50*b*c*d*e*g*(d + e*x) - 12*b^2*e^2*g*(d + e*x) + 3*c^2*e*f*(d + e*x)^2 + 9*c^2*d*g*(d + e*x)^2 - 6*b*c*
e*g*(d + e*x)^2))/(24*e^2*(-2*c*d + b*e)^2*(d + e*x)^(7/2)) + ((-(c^3*e*f) - 3*c^3*d*g + 2*b*c^2*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)))
])/(8*e^2*(2*c*d - b*e)^2*Sqrt[-2*c*d + b*e])

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fricas [B]  time = 0.45, size = 1604, normalized size = 5.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(c^3*d^4*e*f + (c^3*e^5*f + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (3*c^3*d^2*e^3 - 2
*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (3*c^3*d^3*e^2 - 2*b*c^2*d^2*e^3)*g)*x^2 + (3*c^3*d^5 - 2*b*c^2*d^4*
e)*g + 4*(c^3*d^3*e^2*f + (3*c^3*d^4*e - 2*b*c^2*d^3*e^2)*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*((2*c^3*d*e^3 - b*c^2*e^4)*f + (6
*c^3*d^2*e^2 - 7*b*c^2*d*e^3 + 2*b^2*c*e^4)*g)*x^2 - (50*c^3*d^3*e - 85*b*c^2*d^2*e^2 + 46*b^2*c*d*e^3 - 8*b^3
*e^4)*f - (22*c^3*d^4 - 35*b*c^2*d^3*e + 20*b^2*c*d^2*e^2 - 4*b^3*d*e^3)*g + 2*((10*c^3*d^2*e^2 - 7*b*c^2*d*e^
3 + b^2*c*e^4)*f - (34*c^3*d^3*e - 55*b*c^2*d^2*e^2 + 31*b^2*c*d*e^3 - 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(8*c^3*
d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5 + (8*c^3*d^3*e^6 - 12*b*c^2*d^2*e^7 + 6*b^2*c*d*e^8
- b^3*e^9)*x^4 + 4*(8*c^3*d^4*e^5 - 12*b*c^2*d^3*e^6 + 6*b^2*c*d^2*e^7 - b^3*d*e^8)*x^3 + 6*(8*c^3*d^5*e^4 -
12*b*c^2*d^4*e^5 + 6*b^2*c*d^3*e^6 - b^3*d^2*e^7)*x^2 + 4*(8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5
- b^3*d^3*e^6)*x), 1/24*(3*(c^3*d^4*e*f + (c^3*e^5*f + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (
3*c^3*d^2*e^3 - 2*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (3*c^3*d^3*e^2 - 2*b*c^2*d^2*e^3)*g)*x^2 + (3*c^3*d
^5 - 2*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (3*c^3*d^4*e - 2*b*c^2*d^3*e^2)*g)*x)*sqrt(-2*c*d + b*e)*arctan(sqr
t(-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*((2*c^3*d*e^3 - b*c^2*e^4)*f + (6*c^3*d^2*e^2 - 7*b*c^2*d*e^
3 + 2*b^2*c*e^4)*g)*x^2 - (50*c^3*d^3*e - 85*b*c^2*d^2*e^2 + 46*b^2*c*d*e^3 - 8*b^3*e^4)*f - (22*c^3*d^4 - 35*
b*c^2*d^3*e + 20*b^2*c*d^2*e^2 - 4*b^3*d*e^3)*g + 2*((10*c^3*d^2*e^2 - 7*b*c^2*d*e^3 + b^2*c*e^4)*f - (34*c^3*
d^3*e - 55*b*c^2*d^2*e^2 + 31*b^2*c*d*e^3 - 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3
+ 6*b^2*c*d^5*e^4 - b^3*d^4*e^5 + (8*c^3*d^3*e^6 - 12*b*c^2*d^2*e^7 + 6*b^2*c*d*e^8 - b^3*e^9)*x^4 + 4*(8*c^3*
d^4*e^5 - 12*b*c^2*d^3*e^6 + 6*b^2*c*d^2*e^7 - b^3*d*e^8)*x^3 + 6*(8*c^3*d^5*e^4 - 12*b*c^2*d^4*e^5 + 6*b^2*c*
d^3*e^6 - b^3*d^2*e^7)*x^2 + 4*(8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(9/2), x)

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maple [B]  time = 0.11, size = 1033, normalized size = 3.36 \begin {gather*} \frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (6 b \,c^{2} e^{4} g \,x^{3} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-9 c^{3} d \,e^{3} g \,x^{3} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c^{3} e^{4} f \,x^{3} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+18 b \,c^{2} d \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-27 c^{3} d^{2} e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-9 c^{3} d \,e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+18 b \,c^{2} d^{2} e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-27 c^{3} d^{3} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-9 c^{3} d^{2} e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+6 b \,c^{2} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-9 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c^{3} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+9 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}+3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}-12 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} g x +38 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x -2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x -34 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +10 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x -4 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g -8 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} f +12 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,d^{2} e g +30 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f -11 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g -25 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (b e -2 c d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +c d}\, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x)

[Out]

1/24*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g-3*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*e^4*f+38*x*b*c*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c
*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-8*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b
*e-2*c*d)^(1/2)-11*c^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-6*x^2*b*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)+9*x^2*c^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-2*x*b*c*e^3*f*(-c*e*x-b*e+c*d)^(1/
2)*(b*e-2*c*d)^(1/2)-34*x*c^2*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+10*x*c^2*d*e^2*f*(-c*e*x-b*e+c*
d)^(1/2)*(b*e-2*c*d)^(1/2)+12*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+30*b*c*d*e^2*f*(-c*e*x-b*e+
c*d)^(1/2)*(b*e-2*c*d)^(1/2)+18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g+18*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d^2*e^2*g-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))
*x^2*c^3*d^2*e^2*g-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d*e^3*f-27*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^2*e^2*f+6
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g+3*x^2*c^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*
c*d)^(1/2)-12*x*b^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-4*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2
*c*d)^(1/2)-25*c^2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))*x^3*b*c^2*e^4*g-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*d*e^3*g)/(e*x+d)^(7/2)/(b*
e-2*c*d)^(5/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(9/2),x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(9/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**(9/2),x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(9/2), x)

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