3.21.9 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=387 \[ \frac {c^3 (-8 b e g+11 c d g+5 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{7/2}}+\frac {c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)^3}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{96 e^2 (d+e x)^{5/2} (2 c d-b e)^2}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{24 e^2 (d+e x)^{7/2} (2 c d-b e)} \]

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Rubi [A]  time = 0.76, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 6, 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 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)^3}+\frac {c^3 (-8 b e g+11 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 )}{64 e^2 (2 c d-b e)^{7/2}}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{96 e^2 (d+e x)^{5/2} (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}}{4 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+11 c d g+5 c e f)}{24 e^2 (d+e x)^{7/2} (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)^(11/2),x]

[Out]

-((5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(24*e^2*(2*c*d - b*e)*(d + e*x)^(7
/2)) + (c*(5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(96*e^2*(2*c*d - b*e)^2*(d
+ e*x)^(5/2)) + (c^2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*e^2*(2*c*d
- b*e)^3*(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))/(4*e^2*(2*c*d - b*e)*(d
+ e*x)^(11/2)) + (c^3*(5*c*e*f + 11*c*d*g - 8*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])])/(64*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 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)^{11/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c e f+11 c d g-8 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx}{8 e (2 c d-b e)}\\ &=-\frac {(5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {(c (5 c e f+11 c d g-8 b e g)) \int \frac {1}{(d+e x)^{5/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{48 e (2 c d-b e)}\\ &=-\frac {(5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e)^2 (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}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {\left (c^2 (5 c e f+11 c d g-8 b e g)\right ) \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}{64 e (2 c d-b e)^2}\\ &=-\frac {(5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e)^2 (d+e x)^{5/2}}+\frac {c^2 (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e)^3 (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}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {\left (c^3 (5 c e f+11 c d g-8 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}{128 e (2 c d-b e)^3}\\ &=-\frac {(5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e)^2 (d+e x)^{5/2}}+\frac {c^2 (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e)^3 (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}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {\left (c^3 (5 c e f+11 c d g-8 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 )}{64 (2 c d-b e)^3}\\ &=-\frac {(5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 e^2 (2 c d-b e)^2 (d+e x)^{5/2}}+\frac {c^2 (5 c e f+11 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e)^3 (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}}{4 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {c^3 (5 c e f+11 c d g-8 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 )}{64 e^2 (2 c d-b e)^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 128, normalized size = 0.33 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{3/2} \left (-\frac {c^3 (d+e x)^4 (-8 b e g+11 c d g+5 c e f) \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{(b e-2 c d)^4}+3 d g-3 e f\right )}{12 e^2 (d+e x)^{11/2} (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)^(11/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-3*e*f + 3*d*g - (c^3*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^4*Hy
pergeometric2F1[3/2, 4, 5/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(-2*c*d + b*e)^4))/(12*e^2*(2*c*d - b*e)*
(d + e*x)^(11/2))

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IntegrateAlgebraic [A]  time = 2.12, size = 500, normalized size = 1.29 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (-64 b^3 e^3 g (d+e x)+48 b^3 d e^3 g-48 b^3 e^4 f-288 b^2 c d^2 e^2 g-8 b^2 c e^3 f (d+e x)+288 b^2 c d e^3 f+392 b^2 c d e^2 g (d+e x)-16 b^2 c e^2 g (d+e x)^2+576 b c^2 d^3 e g-576 b c^2 d^2 e^2 f-800 b c^2 d^2 e g (d+e x)+32 b c^2 d e^2 f (d+e x)+10 b c^2 e^2 f (d+e x)^2+54 b c^2 d e g (d+e x)^2+24 b c^2 e g (d+e x)^3-384 c^3 d^4 g+384 c^3 d^3 e f+544 c^3 d^3 g (d+e x)-32 c^3 d^2 e f (d+e x)-44 c^3 d^2 g (d+e x)^2-20 c^3 d e f (d+e x)^2-15 c^3 e f (d+e x)^3-33 c^3 d g (d+e x)^3\right )}{192 e^2 (d+e x)^{9/2} (b e-2 c d)^3}+\frac {\left (8 b c^3 e g-11 c^4 d g-5 c^4 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 )}{64 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)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(11/2),x]

[Out]

(Sqrt[2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2]*(384*c^3*d^3*e*f - 576*b*c^2*d^2*e^2*f + 288*b^2*c*d*e^
3*f - 48*b^3*e^4*f - 384*c^3*d^4*g + 576*b*c^2*d^3*e*g - 288*b^2*c*d^2*e^2*g + 48*b^3*d*e^3*g - 32*c^3*d^2*e*f
*(d + e*x) + 32*b*c^2*d*e^2*f*(d + e*x) - 8*b^2*c*e^3*f*(d + e*x) + 544*c^3*d^3*g*(d + e*x) - 800*b*c^2*d^2*e*
g*(d + e*x) + 392*b^2*c*d*e^2*g*(d + e*x) - 64*b^3*e^3*g*(d + e*x) - 20*c^3*d*e*f*(d + e*x)^2 + 10*b*c^2*e^2*f
*(d + e*x)^2 - 44*c^3*d^2*g*(d + e*x)^2 + 54*b*c^2*d*e*g*(d + e*x)^2 - 16*b^2*c*e^2*g*(d + e*x)^2 - 15*c^3*e*f
*(d + e*x)^3 - 33*c^3*d*g*(d + e*x)^3 + 24*b*c^2*e*g*(d + e*x)^3))/(192*e^2*(-2*c*d + b*e)^3*(d + e*x)^(9/2))
+ ((-5*c^4*e*f - 11*c^4*d*g + 8*b*c^3*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)))])/(64*e^2*(2*c*d - b*e)^3*Sqrt[-2*c*d + b*e])

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fricas [B]  time = 0.51, size = 2256, normalized size = 5.83

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)^(11/2),x, algorithm="fricas")

[Out]

[1/384*(3*(5*c^4*d^5*e*f + (5*c^4*e^6*f + (11*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(5*c^4*d*e^5*f + (11*c^4*d^2
*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(5*c^4*d^2*e^4*f + (11*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(5*c^4*d^3
*e^3*f + (11*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (11*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(5*c^4*d^4*e^2*f + (11
*c^4*d^5*e - 8*b*c^3*d^4*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*(5*(2*c^4*d*e^4 - b*c^3*e^5)*f + (22*c^4*d^2*e^3 - 27*b*c^3*d
*e^4 + 8*b^2*c^2*e^5)*g)*x^3 + (5*(26*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 2*b^2*c^2*e^5)*f + (286*c^4*d^3*e^2 - 395
*b*c^3*d^2*e^3 + 158*b^2*c^2*d*e^4 - 16*b^3*c*e^5)*g)*x^2 - (634*c^4*d^4*e - 1385*b*c^3*d^3*e^2 + 1094*b^2*c^2
*d^2*e^3 - 376*b^3*c*d*e^4 + 48*b^4*e^5)*f - (166*c^4*d^5 - 375*b*c^3*d^4*e + 322*b^2*c^2*d^3*e^2 - 120*b^3*c*
d^2*e^3 + 16*b^4*d*e^4)*g + ((234*c^4*d^3*e^2 - 221*b*c^3*d^2*e^3 + 68*b^2*c^2*d*e^4 - 8*b^3*c*e^5)*f - (714*c
^4*d^4*e - 1597*b*c^3*d^3*e^2 + 1340*b^2*c^2*d^2*e^3 - 488*b^3*c*d*e^4 + 64*b^4*e^5)*g)*x)*sqrt(e*x + d))/(16*
c^4*d^9*e^2 - 32*b*c^3*d^8*e^3 + 24*b^2*c^2*d^7*e^4 - 8*b^3*c*d^6*e^5 + b^4*d^5*e^6 + (16*c^4*d^4*e^7 - 32*b*c
^3*d^3*e^8 + 24*b^2*c^2*d^2*e^9 - 8*b^3*c*d*e^10 + b^4*e^11)*x^5 + 5*(16*c^4*d^5*e^6 - 32*b*c^3*d^4*e^7 + 24*b
^2*c^2*d^3*e^8 - 8*b^3*c*d^2*e^9 + b^4*d*e^10)*x^4 + 10*(16*c^4*d^6*e^5 - 32*b*c^3*d^5*e^6 + 24*b^2*c^2*d^4*e^
7 - 8*b^3*c*d^3*e^8 + b^4*d^2*e^9)*x^3 + 10*(16*c^4*d^7*e^4 - 32*b*c^3*d^6*e^5 + 24*b^2*c^2*d^5*e^6 - 8*b^3*c*
d^4*e^7 + b^4*d^3*e^8)*x^2 + 5*(16*c^4*d^8*e^3 - 32*b*c^3*d^7*e^4 + 24*b^2*c^2*d^6*e^5 - 8*b^3*c*d^5*e^6 + b^4
*d^4*e^7)*x), 1/192*(3*(5*c^4*d^5*e*f + (5*c^4*e^6*f + (11*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(5*c^4*d*e^5*f
+ (11*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(5*c^4*d^2*e^4*f + (11*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 +
10*(5*c^4*d^3*e^3*f + (11*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (11*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(5*c^4*d^
4*e^2*f + (11*c^4*d^5*e - 8*b*c^3*d^4*e^2)*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^4*d*e^4 - b*c^3*e^5)*f + (22*c^4*d^2*e^3 - 27*b*c^3*d*e^4 + 8*b^2*c^2*e^5)*g)*x^3 +
(5*(26*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 2*b^2*c^2*e^5)*f + (286*c^4*d^3*e^2 - 395*b*c^3*d^2*e^3 + 158*b^2*c^2*d*
e^4 - 16*b^3*c*e^5)*g)*x^2 - (634*c^4*d^4*e - 1385*b*c^3*d^3*e^2 + 1094*b^2*c^2*d^2*e^3 - 376*b^3*c*d*e^4 + 48
*b^4*e^5)*f - (166*c^4*d^5 - 375*b*c^3*d^4*e + 322*b^2*c^2*d^3*e^2 - 120*b^3*c*d^2*e^3 + 16*b^4*d*e^4)*g + ((2
34*c^4*d^3*e^2 - 221*b*c^3*d^2*e^3 + 68*b^2*c^2*d*e^4 - 8*b^3*c*e^5)*f - (714*c^4*d^4*e - 1597*b*c^3*d^3*e^2 +
1340*b^2*c^2*d^2*e^3 - 488*b^3*c*d*e^4 + 64*b^4*e^5)*g)*x)*sqrt(e*x + d))/(16*c^4*d^9*e^2 - 32*b*c^3*d^8*e^3
+ 24*b^2*c^2*d^7*e^4 - 8*b^3*c*d^6*e^5 + b^4*d^5*e^6 + (16*c^4*d^4*e^7 - 32*b*c^3*d^3*e^8 + 24*b^2*c^2*d^2*e^9
- 8*b^3*c*d*e^10 + b^4*e^11)*x^5 + 5*(16*c^4*d^5*e^6 - 32*b*c^3*d^4*e^7 + 24*b^2*c^2*d^3*e^8 - 8*b^3*c*d^2*e^
9 + b^4*d*e^10)*x^4 + 10*(16*c^4*d^6*e^5 - 32*b*c^3*d^5*e^6 + 24*b^2*c^2*d^4*e^7 - 8*b^3*c*d^3*e^8 + b^4*d^2*e
^9)*x^3 + 10*(16*c^4*d^7*e^4 - 32*b*c^3*d^6*e^5 + 24*b^2*c^2*d^5*e^6 - 8*b^3*c*d^4*e^7 + b^4*d^3*e^8)*x^2 + 5*
(16*c^4*d^8*e^3 - 32*b*c^3*d^7*e^4 + 24*b^2*c^2*d^6*e^5 - 8*b^3*c*d^5*e^6 + b^4*d^4*e^7)*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 {11}{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)^(11/2),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)^(11/2), x)

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maple [B]  time = 0.09, size = 1541, normalized size = 3.98

result too large to display

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)^(11/2),x)

[Out]

-1/192*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^5*g+
15*x^3*c^3*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-126*x^2*b*c^2*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*
c*d)^(1/2)+48*b^3*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-360*x*b^2*c*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b
*e-2*c*d)^(1/2)+620*x*b*c^2*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-52*x*b*c^2*d*e^3*f*(-c*e*x-b*e+
c*d)^(1/2)*(b*e-2*c*d)^(1/2)-83*c^3*d^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-15*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^4*e^5*f-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*e*f+16*b^3*
d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-317*c^3*d^3*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+24*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*b*c^3*e^5*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(
1/2))*x^4*c^4*d*e^4*g-132*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^4*d^2*e^3*g-60*arctan((-c*e*x
-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^4*d*e^4*f-198*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^
4*d^3*e^2*g-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^4*d^2*e^3*f-132*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^4*e*g-60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^3*e^2*f+24*
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^4*e*g+64*x*b^3*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*
d)^(1/2)+96*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^3*d^3*e^2*g+65*x^2*c^3*d*e^3*f*(-c*e*x-b*e+
c*d)^(1/2)*(b*e-2*c*d)^(1/2)+8*x*b^2*c*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-357*x*c^3*d^3*e*g*(-c*e*
x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+117*x*c^3*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-88*b^2*c*d^2*e
^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-280*b^2*c*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+146*b
*c^2*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+534*b*c^2*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(
1/2)-24*x^3*b*c^2*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+33*x^3*c^3*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+16*x^2*b^2*c*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-10*x^2*b*c^2*e^4*f*(-c*e*x-b*e+c*d)
^(1/2)*(b*e-2*c*d)^(1/2)+143*x^2*c^3*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+96*arctan((-c*e*x-b*e+
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^3*d*e^4*g+144*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^3
*d^2*e^3*g)/(e*x+d)^(9/2)/(b*e-2*c*d)^(3/2)/(b^2*e^2-4*b*c*d*e+4*c^2*d^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 {11}{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)^(11/2),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)^(11/2), 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 )}^{11/2}} \,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)^(11/2),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)^(11/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(11/2),x)

[Out]

Timed out

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