3.21.47 \(\int \frac {(d+e x)^{11/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=371 \[ -\frac {32 \sqrt {d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.55, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {32 \sqrt {d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (32*(2*c*d - b*e)^2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d + e*x])/(15*c^5*e^2*Sqrt[d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2]) + (16*(2*c*d - b*e)*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(3/2))/(15*c^4*e^2*Sqrt[d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2]) + (4*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(5/2))/(15*c^3*e^2*Sqrt[d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2]) + (2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(7/2))/(15*c^2*e^2*(2*c*d - b*e)*Sqrt
[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(5 c e f+11 c d g-8 b e g) \int \frac {(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 (5 c e f+11 c d g-8 b e g)) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{5 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(8 (2 c d-b e) (5 c e f+11 c d g-8 b e g)) \int \frac {(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (16 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g)\right ) \int \frac {(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}{15 c^4 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {32 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g) \sqrt {d+e x}}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 263, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {d+e x} \left (128 b^4 e^4 g-16 b^3 c e^3 (47 d g+5 e f-12 e g x)+24 b^2 c^2 e^2 \left (67 d^2 g+3 d e (5 f-13 g x)+e^2 x (2 g x-5 f)\right )-2 b c^3 e \left (741 d^3 g+3 d^2 e (85 f-246 g x)+3 d e^2 x (31 g x-70 f)+e^3 x^2 (15 f+4 g x)\right )+c^4 \left (498 d^4 g+9 d^3 e (25 f-83 g x)+3 d^2 e^2 x (61 g x-115 f)+d e^3 x^2 (75 f+23 g x)+e^4 x^3 (5 f+3 g x)\right )\right )}{15 c^5 e^2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(128*b^4*e^4*g - 16*b^3*c*e^3*(5*e*f + 47*d*g - 12*e*g*x) + 24*b^2*c^2*e^2*(67*d^2*g + 3*d*e*
(5*f - 13*g*x) + e^2*x*(-5*f + 2*g*x)) - 2*b*c^3*e*(741*d^3*g + 3*d^2*e*(85*f - 246*g*x) + e^3*x^2*(15*f + 4*g
*x) + 3*d*e^2*x*(-70*f + 31*g*x)) + c^4*(498*d^4*g + 9*d^3*e*(25*f - 83*g*x) + e^4*x^3*(5*f + 3*g*x) + d*e^3*x
^2*(75*f + 23*g*x) + 3*d^2*e^2*x*(-115*f + 61*g*x))))/(15*c^5*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e
) + c*(d - e*x))])

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IntegrateAlgebraic [A]  time = 7.09, size = 401, normalized size = 1.08 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (128 b^4 e^4 g+192 b^3 c e^3 g (d+e x)-944 b^3 c d e^3 g-80 b^3 c e^4 f+2592 b^2 c^2 d^2 e^2 g-120 b^2 c^2 e^3 f (d+e x)+480 b^2 c^2 d e^3 f+48 b^2 c^2 e^2 g (d+e x)^2-1032 b^2 c^2 d e^2 g (d+e x)-3136 b c^3 d^3 e g-960 b c^3 d^2 e^2 f+1824 b c^3 d^2 e g (d+e x)-30 b c^3 e^2 f (d+e x)^2+480 b c^3 d e^2 f (d+e x)-8 b c^3 e g (d+e x)^3-162 b c^3 d e g (d+e x)^2+1408 c^4 d^4 g+640 c^4 d^3 e f-1056 c^4 d^3 g (d+e x)-480 c^4 d^2 e f (d+e x)+132 c^4 d^2 g (d+e x)^2+5 c^4 e f (d+e x)^3+60 c^4 d e f (d+e x)^2+3 c^4 g (d+e x)^4+11 c^4 d g (d+e x)^3\right )}{15 c^5 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(3/2)*(640*c^4*d^3*e*f - 960*b*c^3*d^2*e^2*f + 480*b^2*c^2*d*e^3*f - 80*b^3*c*e^4*f + 1408*c^4*d
^4*g - 3136*b*c^3*d^3*e*g + 2592*b^2*c^2*d^2*e^2*g - 944*b^3*c*d*e^3*g + 128*b^4*e^4*g - 480*c^4*d^2*e*f*(d +
e*x) + 480*b*c^3*d*e^2*f*(d + e*x) - 120*b^2*c^2*e^3*f*(d + e*x) - 1056*c^4*d^3*g*(d + e*x) + 1824*b*c^3*d^2*e
*g*(d + e*x) - 1032*b^2*c^2*d*e^2*g*(d + e*x) + 192*b^3*c*e^3*g*(d + e*x) + 60*c^4*d*e*f*(d + e*x)^2 - 30*b*c^
3*e^2*f*(d + e*x)^2 + 132*c^4*d^2*g*(d + e*x)^2 - 162*b*c^3*d*e*g*(d + e*x)^2 + 48*b^2*c^2*e^2*g*(d + e*x)^2 +
5*c^4*e*f*(d + e*x)^3 + 11*c^4*d*g*(d + e*x)^3 - 8*b*c^3*e*g*(d + e*x)^3 + 3*c^4*g*(d + e*x)^4))/(15*c^5*e^2*
((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(3/2))

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fricas [A]  time = 0.43, size = 428, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, c^{4} e^{4} g x^{4} + {\left (5 \, c^{4} e^{4} f + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \, {\left (5 \, {\left (5 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (45 \, c^{4} d^{3} e - 102 \, b c^{3} d^{2} e^{2} + 72 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f + 2 \, {\left (249 \, c^{4} d^{4} - 741 \, b c^{3} d^{3} e + 804 \, b^{2} c^{2} d^{2} e^{2} - 376 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g - 3 \, {\left (5 \, {\left (23 \, c^{4} d^{2} e^{2} - 28 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\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 (c^{7} e^{5} x^{3} + c^{7} d^{3} e^{2} - 2 \, b c^{6} d^{2} e^{3} + b^{2} c^{5} d e^{4} - {\left (c^{7} d e^{4} - 2 \, b c^{6} e^{5}\right )} x^{2} - {\left (c^{7} d^{2} e^{3} - b^{2} c^{5} e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/15*(3*c^4*e^4*g*x^4 + (5*c^4*e^4*f + (23*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^3 + 3*(5*(5*c^4*d*e^3 - 2*b*c^3*e^4)
*f + (61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*g)*x^2 + 5*(45*c^4*d^3*e - 102*b*c^3*d^2*e^2 + 72*b^2*
c^2*d*e^3 - 16*b^3*c*e^4)*f + 2*(249*c^4*d^4 - 741*b*c^3*d^3*e + 804*b^2*c^2*d^2*e^2 - 376*b^3*c*d*e^3 + 64*b^
4*e^4)*g - 3*(5*(23*c^4*d^2*e^2 - 28*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*f + (249*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312
*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^7*e^5*x^3 + c
^7*d^3*e^2 - 2*b*c^6*d^2*e^3 + b^2*c^5*d*e^4 - (c^7*d*e^4 - 2*b*c^6*e^5)*x^2 - (c^7*d^2*e^3 - b^2*c^5*e^5)*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((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 367, normalized size = 0.99 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{4} x^{4} c^{4}-8 b \,c^{3} e^{4} g \,x^{3}+23 c^{4} d \,e^{3} g \,x^{3}+5 c^{4} e^{4} f \,x^{3}+48 b^{2} c^{2} e^{4} g \,x^{2}-186 b \,c^{3} d \,e^{3} g \,x^{2}-30 b \,c^{3} e^{4} f \,x^{2}+183 c^{4} d^{2} e^{2} g \,x^{2}+75 c^{4} d \,e^{3} f \,x^{2}+192 b^{3} c \,e^{4} g x -936 b^{2} c^{2} d \,e^{3} g x -120 b^{2} c^{2} e^{4} f x +1476 b \,c^{3} d^{2} e^{2} g x +420 b \,c^{3} d \,e^{3} f x -747 c^{4} d^{3} e g x -345 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -752 b^{3} c d \,e^{3} g -80 b^{3} c \,e^{4} f +1608 b^{2} c^{2} d^{2} e^{2} g +360 b^{2} c^{2} d \,e^{3} f -1482 b \,c^{3} d^{3} e g -510 b \,c^{3} d^{2} e^{2} f +498 c^{4} d^{4} g +225 c^{4} d^{3} e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{15 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} c^{5} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/15*(c*e*x+b*e-c*d)*(3*c^4*e^4*g*x^4-8*b*c^3*e^4*g*x^3+23*c^4*d*e^3*g*x^3+5*c^4*e^4*f*x^3+48*b^2*c^2*e^4*g*x^
2-186*b*c^3*d*e^3*g*x^2-30*b*c^3*e^4*f*x^2+183*c^4*d^2*e^2*g*x^2+75*c^4*d*e^3*f*x^2+192*b^3*c*e^4*g*x-936*b^2*
c^2*d*e^3*g*x-120*b^2*c^2*e^4*f*x+1476*b*c^3*d^2*e^2*g*x+420*b*c^3*d*e^3*f*x-747*c^4*d^3*e*g*x-345*c^4*d^2*e^2
*f*x+128*b^4*e^4*g-752*b^3*c*d*e^3*g-80*b^3*c*e^4*f+1608*b^2*c^2*d^2*e^2*g+360*b^2*c^2*d*e^3*f-1482*b*c^3*d^3*
e*g-510*b*c^3*d^2*e^2*f+498*c^4*d^4*g+225*c^4*d^3*e*f)*(e*x+d)^(5/2)/c^5/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(5/2)

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maxima [A]  time = 0.97, size = 363, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (c^{3} e^{3} x^{3} + 45 \, c^{3} d^{3} - 102 \, b c^{2} d^{2} e + 72 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 3 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{3 \, {\left (c^{5} e^{2} x - c^{5} d e + b c^{4} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 498 \, c^{4} d^{4} - 1482 \, b c^{3} d^{3} e + 1608 \, b^{2} c^{2} d^{2} e^{2} - 752 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} - 3 \, {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{15 \, {\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt {-c e x + c d - b e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/3*(c^3*e^3*x^3 + 45*c^3*d^3 - 102*b*c^2*d^2*e + 72*b^2*c*d*e^2 - 16*b^3*e^3 + 3*(5*c^3*d*e^2 - 2*b*c^2*e^3)*
x^2 - 3*(23*c^3*d^2*e - 28*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*f/((c^5*e^2*x - c^5*d*e + b*c^4*e^2)*sqrt(-c*e*x + c*
d - b*e)) + 2/15*(3*c^4*e^4*x^4 + 498*c^4*d^4 - 1482*b*c^3*d^3*e + 1608*b^2*c^2*d^2*e^2 - 752*b^3*c*d*e^3 + 12
8*b^4*e^4 + (23*c^4*d*e^3 - 8*b*c^3*e^4)*x^3 + 3*(61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 - 3*(2
49*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/((c^6*e^3*x - c^6*d*e^2 + b*c^5*e^3)
*sqrt(-c*e*x + c*d - b*e))

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mupad [B]  time = 3.29, size = 435, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1504\,g\,b^3\,c\,d\,e^3-160\,f\,b^3\,c\,e^4+3216\,g\,b^2\,c^2\,d^2\,e^2+720\,f\,b^2\,c^2\,d\,e^3-2964\,g\,b\,c^3\,d^3\,e-1020\,f\,b\,c^3\,d^2\,e^2+996\,g\,c^4\,d^4+450\,f\,c^4\,d^3\,e\right )}{15\,c^7\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-62\,g\,b\,c\,d\,e-10\,f\,b\,c\,e^2+61\,g\,c^2\,d^2+25\,f\,c^2\,d\,e\right )}{5\,c^5\,e^3}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (23\,c\,d\,g-8\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^4\,e^2}+\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{5\,c^3\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (-384\,g\,b^3\,c\,e^4+1872\,g\,b^2\,c^2\,d\,e^3+240\,f\,b^2\,c^2\,e^4-2952\,g\,b\,c^3\,d^2\,e^2-840\,f\,b\,c^3\,d\,e^3+1494\,g\,c^4\,d^3\,e+690\,f\,c^4\,d^2\,e^2\right )}{15\,c^7\,e^5}\right )}{x^3+\frac {x\,\left (15\,b^2\,c^5\,e^5-15\,c^7\,d^2\,e^3\right )}{15\,c^7\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^(11/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(256*b^4*e^4*g + 996*c^4*d^4*g - 160*b^3*c*e^4
*f + 450*c^4*d^3*e*f - 2964*b*c^3*d^3*e*g - 1504*b^3*c*d*e^3*g - 1020*b*c^3*d^2*e^2*f + 720*b^2*c^2*d*e^3*f +
3216*b^2*c^2*d^2*e^2*g))/(15*c^7*e^5) + (2*x^2*(d + e*x)^(1/2)*(16*b^2*e^2*g + 61*c^2*d^2*g - 10*b*c*e^2*f + 2
5*c^2*d*e*f - 62*b*c*d*e*g))/(5*c^5*e^3) + (2*x^3*(d + e*x)^(1/2)*(23*c*d*g - 8*b*e*g + 5*c*e*f))/(15*c^4*e^2)
+ (2*g*x^4*(d + e*x)^(1/2))/(5*c^3*e) - (x*(d + e*x)^(1/2)*(240*b^2*c^2*e^4*f + 690*c^4*d^2*e^2*f - 384*b^3*c
*e^4*g + 1494*c^4*d^3*e*g - 840*b*c^3*d*e^3*f - 2952*b*c^3*d^2*e^2*g + 1872*b^2*c^2*d*e^3*g))/(15*c^7*e^5)))/(
x^3 + (x*(15*b^2*c^5*e^5 - 15*c^7*d^2*e^3))/(15*c^7*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x^2*(2*b*e - c*d))/(
c*e))

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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((e*x+d)**(11/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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