3.21.27 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=360 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}}+\frac {(2 c d-b e)^{3/2} (2 b e g-9 c d g+5 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2} \]

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Rubi [A]  time = 0.61, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {792, 664, 660, 208} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}}+\frac {(2 c d-b e)^{3/2} (2 b e g-9 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 )}{e^2} \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)^(9/2),x]

[Out]

-(((2*c*d - b*e)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqrt[d + e*x]))
- ((5*c*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) - ((5*c
*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) -
((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) + ((2*c*d - b*e
)^(3/2)*(5*c*e*f - 9*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]*Sqr
t[d + e*x])])/e^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 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)^{9/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(5 c e f-9 c d g+2 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)^{7/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 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 )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {((-2 c d+b e) (5 c e f-9 c d g+2 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)^{5/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(5 c e f-9 c d g+2 b e 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)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 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 )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {((2 c d-b e) (5 c e f-9 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)^{3/2}} \, dx}{2 e}\\ &=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e 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)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 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 )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {\left ((2 c d-b e)^2 (5 c e f-9 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}{2 e}\\ &=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e 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)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 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 )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\left ((2 c d-b e)^2 (5 c e f-9 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 )\\ &=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e 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)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 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 )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 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 )}{e^2}\\ \end {align*}

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Mathematica [A]  time = 0.66, size = 221, normalized size = 0.61 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{5/2} \left ((e f-d g) (b e-c d+c e x)-\frac {(d+e x) (2 b e g-9 c d g+5 c e f) \left (\sqrt {c (d-e x)-b e} \left (23 b^2 e^2+b c e (11 e x-81 d)+c^2 \left (73 d^2-16 d e x+3 e^2 x^2\right )\right )-15 (2 c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{15 (c (d-e x)-b e)^{5/2}}\right )}{e^2 (d+e x)^{7/2} (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)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((e*f - d*g)*(-(c*d) + b*e + c*e*x) - ((5*c*e*f - 9*c*d*g + 2*b*e*g)
*(d + e*x)*(Sqrt[-(b*e) + c*(d - e*x)]*(23*b^2*e^2 + b*c*e*(-81*d + 11*e*x) + c^2*(73*d^2 - 16*d*e*x + 3*e^2*x
^2)) - 15*(2*c*d - b*e)^(5/2)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]]))/(15*(-(b*e) + c*(d - e*x))^
(5/2))))/(e^2*(2*c*d - b*e)*(d + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 4.76, size = 382, normalized size = 1.06 \begin {gather*} \frac {\sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (46 b^2 e^2 g (d+e x)+15 b^2 d e^2 g-15 b^2 e^3 f-60 b c d^2 e g+70 b c e^2 f (d+e x)+60 b c d e^2 f-254 b c d e g (d+e x)+22 b c e g (d+e x)^2+60 c^2 d^3 g-60 c^2 d^2 e f+324 c^2 d^2 g (d+e x)-140 c^2 d e f (d+e x)+10 c^2 e f (d+e x)^2-54 c^2 d g (d+e x)^2+6 c^2 g (d+e x)^3\right )}{15 e^2 (d+e x)^{3/2}}+\frac {\left (-2 b^3 e^3 g+17 b^2 c d e^2 g-5 b^2 c e^3 f-44 b c^2 d^2 e g+20 b c^2 d e^2 f+36 c^3 d^3 g-20 c^3 d^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 )}{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)^(9/2),x]

[Out]

(Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]*(-60*c^2*d^2*e*f + 60*b*c*d*e^2*f - 15*b^2*e^3*f + 60*c^2*d^3*g
- 60*b*c*d^2*e*g + 15*b^2*d*e^2*g - 140*c^2*d*e*f*(d + e*x) + 70*b*c*e^2*f*(d + e*x) + 324*c^2*d^2*g*(d + e*x
) - 254*b*c*d*e*g*(d + e*x) + 46*b^2*e^2*g*(d + e*x) + 10*c^2*e*f*(d + e*x)^2 - 54*c^2*d*g*(d + e*x)^2 + 22*b*
c*e*g*(d + e*x)^2 + 6*c^2*g*(d + e*x)^3))/(15*e^2*(d + e*x)^(3/2)) + ((-20*c^3*d^2*e*f + 20*b*c^2*d*e^2*f - 5*
b^2*c*e^3*f + 36*c^3*d^3*g - 44*b*c^2*d^2*e*g + 17*b^2*c*d*e^2*g - 2*b^3*e^3*g)*ArcTan[(Sqrt[-2*c*d + b*e]*Sqr
t[(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)))])/(e^2*Sqrt[-2*c*d +
b*e])

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fricas [A]  time = 0.43, size = 990, normalized size = 2.75 \begin {gather*} \left [-\frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} 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}}\right ) - 2 \, {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{30 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\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}\right ) + {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \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)^(9/2),x, algorithm="fricas")

[Out]

[-1/30*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e
- b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c
^2*d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*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*(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*
c*d*e^2 + 3*b^2*e^3)*f + (336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f -
(117*c^2*d^2*e - 105*b*c*d*e^2 + 23*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(
e^4*x^2 + 2*d*e^3*x + d^2*e^2), 1/15*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^
2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*
d^2*e^2 - b*c*d*e^3)*f - (18*c^2*d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*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)) +
(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^
2*e^3)*f + (336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f - (117*c^2*d^2*e
- 105*b*c*d*e^2 + 23*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(e^4*x^2 + 2*d*
e^3*x + d^2*e^2)]

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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)^(9/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.08, size = 1136, normalized size = 3.16 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (30 b^{3} e^{4} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-255 b^{2} c d \,e^{3} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+75 b^{2} c \,e^{4} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 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 )-300 b \,c^{2} d \,e^{3} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-540 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 )+300 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 )+30 b^{3} d \,e^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-255 b^{2} c \,d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+75 b^{2} c d \,e^{3} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 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 )-300 b \,c^{2} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-540 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+300 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}\, c^{2} e^{3} g \,x^{3}-22 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+36 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-10 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}-46 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} g x +210 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x -70 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x -234 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +120 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x -61 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +15 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} f +292 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,d^{2} e g -130 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f -336 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g +190 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{15 \left (e x +d \right )^{\frac {3}{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)^(9/2),x)

[Out]

-1/15*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-540*c^3*d^4*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+
30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^3*e^4*g+210*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b
*c*d*e^2*g*x+300*c^3*d^3*e*f*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+15*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2
*c*d)^(1/2)*b^2*e^3*f-336*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^3*g+30*arctan((-c*e*x-b*e+c*d)^(1/2)/
(b*e-2*c*d)^(1/2))*b^3*d*e^3*g-22*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*e^3*g*x^2+36*(-c*e*x-b*e+c*d)^(
1/2)*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2-70*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x-234*(-c*e*x-b*e
+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^2*e*g*x+120*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x-130*(-c
*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f+660*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))+75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*f-300*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f-6*x^3*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+75*arctan((-c*e*x
-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^2*c*e^4*f-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d^
2*e^2*g-540*c^3*d^3*e*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+300*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))+660*b*c^2*d^3*e*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-10*(-c*e*
x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*e^3*f*x^2-46*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x-61*(-
c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+190*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f-30
0*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d*e^3*f-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*
d)^(1/2))*x*b^2*c*d*e^3*g+292*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(3/2)/(-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 {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)^(5/2)/(e*x+d)^(9/2),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)^(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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\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)^(5/2))/(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)^(5/2))/(d + e*x)^(9/2), 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)**(9/2),x)

[Out]

Timed out

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