3.21.29 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=383 \[ -\frac {5 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (6 b e g-13 c d g+c e f)}{8 e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {5 c^2 (6 b e g-13 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{8 e^2 \sqrt {2 c d-b e}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (6 b e g-13 c d g+c e f)}{12 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (6 b e g-13 c d g+c e f)}{24 e^2 (d+e x)^{5/2} (2 c d-b e)} \]

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Rubi [A]  time = 0.59, antiderivative size = 383, 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, 664, 660, 208} \begin {gather*} -\frac {5 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (6 b e g-13 c d g+c e f)}{8 e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {5 c^2 (6 b e g-13 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 \sqrt {2 c d-b e}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^{13/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (6 b e g-13 c d g+c e f)}{12 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (6 b e g-13 c d g+c e f)}{24 e^2 (d+e x)^{5/2} (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(-5*c^2*(c*e*f - 13*c*d*g + 6*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)*Sqrt[d +
e*x]) - (5*c*(c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2*(2*c*d - b*e)*(
d + e*x)^(5/2)) + ((c*e*f - 13*c*d*g + 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(12*e^2*(2*c*d -
b*e)*(d + e*x)^(9/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*
x)^(13/2)) + (5*c^2*(c*e*f - 13*c*d*g + 6*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*Sqrt[2*c*d - b*e])

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 664

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 + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), 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] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {(c e f-13 c d g+6 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx}{6 e (2 c d-b e)}\\ &=\frac {(c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {(5 c (c e f-13 c d g+6 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{24 e (2 c d-b e)}\\ &=-\frac {5 c (c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {\left (5 c^2 (c e f-13 c d g+6 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{16 e (2 c d-b e)}\\ &=-\frac {5 c^2 (c e f-13 c d g+6 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) \sqrt {d+e x}}-\frac {5 c (c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {\left (5 c^2 (c e f-13 c d g+6 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}\\ &=-\frac {5 c^2 (c e f-13 c d g+6 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) \sqrt {d+e x}}-\frac {5 c (c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {1}{8} \left (5 c^2 (c e f-13 c d g+6 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 )\\ &=-\frac {5 c^2 (c e f-13 c d g+6 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) \sqrt {d+e x}}-\frac {5 c (c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-13 c d g+6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {5 c^2 (c e f-13 c d g+6 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 \sqrt {2 c d-b e}}\\ \end {align*}

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Mathematica [C]  time = 0.14, size = 127, normalized size = 0.33 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{7/2} \left (\frac {c^2 (d+e x)^3 (6 b e g-13 c d g+c e f) \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{(2 c d-b e)^3}+7 d g-7 e f\right )}{21 e^2 (d+e x)^{13/2} (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(7/2)*(-7*e*f + 7*d*g + (c^2*(c*e*f - 13*c*d*g + 6*b*e*g)*(d + e*x)^3*Hype
rgeometric2F1[3, 7/2, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(2*c*d - b*e)^3))/(21*e^2*(2*c*d - b*e)*(d
+ e*x)^(13/2))

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IntegrateAlgebraic [A]  time = 5.43, size = 339, normalized size = 0.89 \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-26 b c e^2 f (d+e x)+32 b c d e^2 f+74 b c d e g (d+e x)-54 b c e g (d+e x)^2+32 c^2 d^3 g-32 c^2 d^2 e f-100 c^2 d^2 g (d+e x)+52 c^2 d e f (d+e x)-33 c^2 e f (d+e x)^2+141 c^2 d g (d+e x)^2+48 c^2 g (d+e x)^3\right )}{24 e^2 (d+e x)^{7/2}}+\frac {5 \left (-6 b c^2 e g+13 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 \sqrt {b e-2 c d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(13/2),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 + 52*c^2*d*e*f*(d + e*x) - 26*b*c*e^2*f*(d + e*x) - 100*c^2*d^2*g*(d
+ e*x) + 74*b*c*d*e*g*(d + e*x) - 12*b^2*e^2*g*(d + e*x) - 33*c^2*e*f*(d + e*x)^2 + 141*c^2*d*g*(d + e*x)^2 -
54*b*c*e*g*(d + e*x)^2 + 48*c^2*g*(d + e*x)^3))/(24*e^2*(d + e*x)^(7/2)) + (5*(-(c^3*e*f) + 13*c^3*d*g - 6*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*Sqrt[-2*c*d + b*e])

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fricas [A]  time = 0.51, size = 1388, normalized size = 3.62

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)^(5/2)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

[1/48*(15*(c^3*d^4*e*f + (c^3*e^5*f - (13*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f - (13*c^3*d^2*e^3 -
6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f - (13*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x^2 - (13*c^3*d^5 - 6*b*c^2*
d^4*e)*g + 4*(c^3*d^3*e^2*f - (13*c^3*d^4*e - 6*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)*(48*(2*c^3*d*e^3 - b*c^2*e^4)*g
*x^3 - 3*(11*(2*c^3*d*e^3 - b*c^2*e^4)*f - (190*c^3*d^2*e^2 - 131*b*c^2*d*e^3 + 18*b^2*c*e^4)*g)*x^2 - (26*c^3
*d^3*e - 25*b*c^2*d^2*e^2 + 22*b^2*c*d*e^3 - 8*b^3*e^4)*f + (242*c^3*d^4 - 145*b*c^2*d^3*e + 4*b^2*c*d^2*e^2 +
4*b^3*d*e^3)*g - 2*((14*c^3*d^2*e^2 + 19*b*c^2*d*e^3 - 13*b^2*c*e^4)*f - (326*c^3*d^3*e - 197*b*c^2*d^2*e^2 +
5*b^2*c*d*e^3 + 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(2*c*d^5*e^2 - b*d^4*e^3 + (2*c*d*e^6 - b*e^7)*x^4 + 4*(2*c*d
^2*e^5 - b*d*e^6)*x^3 + 6*(2*c*d^3*e^4 - b*d^2*e^5)*x^2 + 4*(2*c*d^4*e^3 - b*d^3*e^4)*x), 1/24*(15*(c^3*d^4*e*
f + (c^3*e^5*f - (13*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f - (13*c^3*d^2*e^3 - 6*b*c^2*d*e^4)*g)*x^
3 + 6*(c^3*d^2*e^3*f - (13*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x^2 - (13*c^3*d^5 - 6*b*c^2*d^4*e)*g + 4*(c^3*d^3
*e^2*f - (13*c^3*d^4*e - 6*b*c^2*d^3*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)*(48*(2*c^3*d*e^3 - b*c^2*e^4)*g*x^3 - 3*(11*(2*c^3*d*e^3 - b*c^2*e^4)*f - (190*c^3*d^2*e^2 - 131
*b*c^2*d*e^3 + 18*b^2*c*e^4)*g)*x^2 - (26*c^3*d^3*e - 25*b*c^2*d^2*e^2 + 22*b^2*c*d*e^3 - 8*b^3*e^4)*f + (242*
c^3*d^4 - 145*b*c^2*d^3*e + 4*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g - 2*((14*c^3*d^2*e^2 + 19*b*c^2*d*e^3 - 13*b^2*c*
e^4)*f - (326*c^3*d^3*e - 197*b*c^2*d^2*e^2 + 5*b^2*c*d*e^3 + 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(2*c*d^5*e^2 - b
*d^4*e^3 + (2*c*d*e^6 - b*e^7)*x^4 + 4*(2*c*d^2*e^5 - b*d*e^6)*x^3 + 6*(2*c*d^3*e^4 - b*d^2*e^5)*x^2 + 4*(2*c*
d^4*e^3 - b*d^3*e^4)*x)]

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giac [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)^(5/2)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.09, size = 1070, normalized size = 2.79 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (90 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 )-195 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 )+15 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 )+270 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 )-585 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 )+45 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 )+270 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 )-585 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 )+45 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 )+90 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 )-195 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+15 c^{3} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-48 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} g \,x^{3}+54 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}-285 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}+33 \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 +34 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x +26 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x -326 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +14 \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 -6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f -121 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g +13 \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}} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 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)^(5/2)/(e*x+d)^(13/2),x)

[Out]

-1/24*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-195*c^3*d^4*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+
15*c^3*e^4*f*x^3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+34*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*
b*c*d*e^2*g*x+15*c^3*d^3*e*f*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+8*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*
c*d)^(1/2)*b^2*e^3*f-121*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^3*g+54*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c
*d)^(1/2)*b*c*e^3*g*x^2-285*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2+26*(-c*e*x-b*e+c*d)^(1/2)
*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x-326*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^2*e*g*x+14*(-c*e*x-b*e+c*d)^
(1/2)*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x-6*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f+270*b*c^2*d*e^3*g
*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+270*b*c^2*d^2*e^2*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))-48*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*e^3*g*x^3-585*c^3*d^2*e^2*g*x^2*arctan((-c*e*x-
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+45*c^3*d*e^3*f*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-585*c^3*
d^3*e*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+45*c^3*d^2*e^2*f*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))+90*b*c^2*d^3*e*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+33*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)*c^2*e^3*f*x^2+12*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x+4*(-c*e*x-b*e+c*d)^(1/2
)*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+13*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f+90*b*c^2*e^4*g*x^3*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-195*c^3*d*e^3*g*x^3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))+12*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*d^2*e*g)/(e*x+d)^(7/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*
c*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {13}{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)^(5/2)/(e*x+d)^(13/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(13/2), 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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{13/2}} \,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)^(5/2))/(d + e*x)^(13/2),x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(13/2), 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)**(5/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

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