3.2276 \(\int \frac{(d+e x)^{13/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=448 \[ \frac{2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{256 \sqrt{d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(13/2))/(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
)) - (256*(2*c*d - b*e)^3*(7*c*e*f + 13*c*d*g - 10*b*e*g)*Sqrt[d + e*x])/(105*c^6*e^2*Sqrt[d*(c*d - b*e) - b*e
^2*x - c*e^2*x^2]) + (128*(2*c*d - b*e)^2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(3/2))/(105*c^5*e^2*Sqrt[d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (32*(2*c*d - b*e)*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(5/2))/(105
*c^4*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (16*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(7/2))/(10
5*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(9/2))/(21
*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

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Rubi [A]  time = 0.705072, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{256 \sqrt{d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(13/2)*(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)^(13/2))/(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
)) - (256*(2*c*d - b*e)^3*(7*c*e*f + 13*c*d*g - 10*b*e*g)*Sqrt[d + e*x])/(105*c^6*e^2*Sqrt[d*(c*d - b*e) - b*e
^2*x - c*e^2*x^2]) + (128*(2*c*d - b*e)^2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(3/2))/(105*c^5*e^2*Sqrt[d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (32*(2*c*d - b*e)*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(5/2))/(105
*c^4*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (16*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(7/2))/(10
5*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(9/2))/(21
*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 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]

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 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]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{13/2} (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)^{13/2}}{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{(7 c e f+13 c d g-10 b e g) \int \frac{(d+e x)^{11/2}}{\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)^{13/2}}{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 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 (7 c e f+13 c d g-10 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}{21 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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{(16 (2 c d-b e) (7 c e f+13 c d g-10 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}{35 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 (64 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g)\right ) \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}{105 c^4 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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{128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 (128 (2 c d-b e)^3 (7 c e f+13 c d g-10 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}{105 c^5 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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{256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt{d+e x}}{105 c^6 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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*}

Mathematica [A]  time = 0.350648, size = 366, normalized size = 0.82 \[ \frac{2 \sqrt{d+e x} \left (16 b^2 c^3 e^2 \left (3 d^2 e (287 f-681 g x)+2844 d^3 g+6 d e^2 x (29 g x-77 f)+e^3 x^2 (21 f+5 g x)\right )-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (5 g x-14 f)\right )+128 b^4 c e^4 (78 d g+7 e f-15 e g x)-1280 b^5 e^5 g-2 b c^4 e \left (6 d^2 e^2 x (449 g x-1106 f)+12 d^3 e (581 f-1482 g x)+16563 d^4 g+12 d e^3 x^2 (63 f+16 g x)+e^4 x^3 (28 f+15 g x)\right )+c^5 \left (2 d^2 e^3 x^2 (903 f+257 g x)+12 d^3 e^2 x (292 g x-637 f)+3 d^4 e (1687 f-4707 g x)+9414 d^5 g+2 d e^4 x^3 (98 f+57 g x)+3 e^5 x^4 (7 f+5 g x)\right )\right )}{105 c^6 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(13/2)*(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]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(7*e*f + 78*d*g - 15*e*g*x) - 32*b^3*c^2*e^3*(953*d^2*g + 2*
d*e*(91*f - 204*g*x) + 3*e^2*x*(-14*f + 5*g*x)) + 16*b^2*c^3*e^2*(2844*d^3*g + 3*d^2*e*(287*f - 681*g*x) + e^3
*x^2*(21*f + 5*g*x) + 6*d*e^2*x*(-77*f + 29*g*x)) + c^5*(9414*d^5*g + 3*d^4*e*(1687*f - 4707*g*x) + 3*e^5*x^4*
(7*f + 5*g*x) + 2*d*e^4*x^3*(98*f + 57*g*x) + 2*d^2*e^3*x^2*(903*f + 257*g*x) + 12*d^3*e^2*x*(-637*f + 292*g*x
)) - 2*b*c^4*e*(16563*d^4*g + 12*d^3*e*(581*f - 1482*g*x) + e^4*x^3*(28*f + 15*g*x) + 12*d*e^3*x^2*(63*f + 16*
g*x) + 6*d^2*e^2*x*(-1106*f + 449*g*x))))/(105*c^6*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d -
e*x))])

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Maple [A]  time = 0.007, size = 535, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -15\,g{e}^{5}{x}^{5}{c}^{5}+30\,b{c}^{4}{e}^{5}g{x}^{4}-114\,{c}^{5}d{e}^{4}g{x}^{4}-21\,{c}^{5}{e}^{5}f{x}^{4}-80\,{b}^{2}{c}^{3}{e}^{5}g{x}^{3}+384\,b{c}^{4}d{e}^{4}g{x}^{3}+56\,b{c}^{4}{e}^{5}f{x}^{3}-514\,{c}^{5}{d}^{2}{e}^{3}g{x}^{3}-196\,{c}^{5}d{e}^{4}f{x}^{3}+480\,{b}^{3}{c}^{2}{e}^{5}g{x}^{2}-2784\,{b}^{2}{c}^{3}d{e}^{4}g{x}^{2}-336\,{b}^{2}{c}^{3}{e}^{5}f{x}^{2}+5388\,b{c}^{4}{d}^{2}{e}^{3}g{x}^{2}+1512\,b{c}^{4}d{e}^{4}f{x}^{2}-3504\,{c}^{5}{d}^{3}{e}^{2}g{x}^{2}-1806\,{c}^{5}{d}^{2}{e}^{3}f{x}^{2}+1920\,{b}^{4}c{e}^{5}gx-13056\,{b}^{3}{c}^{2}d{e}^{4}gx-1344\,{b}^{3}{c}^{2}{e}^{5}fx+32688\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}gx+7392\,{b}^{2}{c}^{3}d{e}^{4}fx-35568\,b{c}^{4}{d}^{3}{e}^{2}gx-13272\,b{c}^{4}{d}^{2}{e}^{3}fx+14121\,{c}^{5}{d}^{4}egx+7644\,{c}^{5}{d}^{3}{e}^{2}fx+1280\,{b}^{5}{e}^{5}g-9984\,{b}^{4}cd{e}^{4}g-896\,{b}^{4}c{e}^{5}f+30496\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}g+5824\,{b}^{3}{c}^{2}d{e}^{4}f-45504\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}g-13776\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}f+33126\,b{c}^{4}{d}^{4}eg+13944\,b{c}^{4}{d}^{3}{e}^{2}f-9414\,{c}^{5}{d}^{5}g-5061\,f{d}^{4}{c}^{5}e \right ) }{105\,{c}^{6}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(c*e*x+b*e-c*d)*(-15*c^5*e^5*g*x^5+30*b*c^4*e^5*g*x^4-114*c^5*d*e^4*g*x^4-21*c^5*e^5*f*x^4-80*b^2*c^3*e
^5*g*x^3+384*b*c^4*d*e^4*g*x^3+56*b*c^4*e^5*f*x^3-514*c^5*d^2*e^3*g*x^3-196*c^5*d*e^4*f*x^3+480*b^3*c^2*e^5*g*
x^2-2784*b^2*c^3*d*e^4*g*x^2-336*b^2*c^3*e^5*f*x^2+5388*b*c^4*d^2*e^3*g*x^2+1512*b*c^4*d*e^4*f*x^2-3504*c^5*d^
3*e^2*g*x^2-1806*c^5*d^2*e^3*f*x^2+1920*b^4*c*e^5*g*x-13056*b^3*c^2*d*e^4*g*x-1344*b^3*c^2*e^5*f*x+32688*b^2*c
^3*d^2*e^3*g*x+7392*b^2*c^3*d*e^4*f*x-35568*b*c^4*d^3*e^2*g*x-13272*b*c^4*d^2*e^3*f*x+14121*c^5*d^4*e*g*x+7644
*c^5*d^3*e^2*f*x+1280*b^5*e^5*g-9984*b^4*c*d*e^4*g-896*b^4*c*e^5*f+30496*b^3*c^2*d^2*e^3*g+5824*b^3*c^2*d*e^4*
f-45504*b^2*c^3*d^3*e^2*g-13776*b^2*c^3*d^2*e^3*f+33126*b*c^4*d^4*e*g+13944*b*c^4*d^3*e^2*f-9414*c^5*d^5*g-506
1*c^5*d^4*e*f)*(e*x+d)^(5/2)/c^6/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 = 1.44527, size = 690, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \,{\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \,{\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \,{\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \,{\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \,{\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \,{\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \,{\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \,{\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \,{\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt{-c e x + c d - b e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*c^4*e^4*x^4 + 723*c^4*d^4 - 1992*b*c^3*d^3*e + 1968*b^2*c^2*d^2*e^2 - 832*b^3*c*d*e^3 + 128*b^4*e^4 +
4*(7*c^4*d*e^3 - 2*b*c^3*e^4)*x^3 + 6*(43*c^4*d^2*e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*x^2 - 12*(91*c^4*d^3*e
- 158*b*c^3*d^2*e^2 + 88*b^2*c^2*d*e^3 - 16*b^3*c*e^4)*x)*f/((c^6*e^2*x - c^6*d*e + b*c^5*e^2)*sqrt(-c*e*x +
c*d - b*e)) + 2/105*(15*c^5*e^5*x^5 + 9414*c^5*d^5 - 33126*b*c^4*d^4*e + 45504*b^2*c^3*d^3*e^2 - 30496*b^3*c^2
*d^2*e^3 + 9984*b^4*c*d*e^4 - 1280*b^5*e^5 + 6*(19*c^5*d*e^4 - 5*b*c^4*e^5)*x^4 + 2*(257*c^5*d^2*e^3 - 192*b*c
^4*d*e^4 + 40*b^2*c^3*e^5)*x^3 + 12*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e^4 - 40*b^3*c^2*e^5)
*x^2 - 3*(4707*c^5*d^4*e - 11856*b*c^4*d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*x
)*g/((c^7*e^3*x - c^7*d*e^2 + b*c^6*e^3)*sqrt(-c*e*x + c*d - b*e))

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Fricas [A]  time = 1.52373, size = 1250, normalized size = 2.79 \begin{align*} -\frac{2 \,{\left (15 \, c^{5} e^{5} g x^{5} + 3 \,{\left (7 \, c^{5} e^{5} f + 2 \,{\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 2 \,{\left (14 \,{\left (7 \, c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} f +{\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 6 \,{\left (7 \,{\left (43 \, c^{5} d^{2} e^{3} - 36 \, b c^{4} d e^{4} + 8 \, b^{2} c^{3} e^{5}\right )} f + 2 \,{\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 7 \,{\left (723 \, c^{5} d^{4} e - 1992 \, b c^{4} d^{3} e^{2} + 1968 \, b^{2} c^{3} d^{2} e^{3} - 832 \, b^{3} c^{2} d e^{4} + 128 \, b^{4} c e^{5}\right )} f + 2 \,{\left (4707 \, c^{5} d^{5} - 16563 \, b c^{4} d^{4} e + 22752 \, b^{2} c^{3} d^{3} e^{2} - 15248 \, b^{3} c^{2} d^{2} e^{3} + 4992 \, b^{4} c d e^{4} - 640 \, b^{5} e^{5}\right )} g - 3 \,{\left (28 \,{\left (91 \, c^{5} d^{3} e^{2} - 158 \, b c^{4} d^{2} e^{3} + 88 \, b^{2} c^{3} d e^{4} - 16 \, b^{3} c^{2} e^{5}\right )} f +{\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\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}}{105 \,{\left (c^{8} e^{5} x^{3} + c^{8} d^{3} e^{2} - 2 \, b c^{7} d^{2} e^{3} + b^{2} c^{6} d e^{4} -{\left (c^{8} d e^{4} - 2 \, b c^{7} e^{5}\right )} x^{2} -{\left (c^{8} d^{2} e^{3} - b^{2} c^{6} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(15*c^5*e^5*g*x^5 + 3*(7*c^5*e^5*f + 2*(19*c^5*d*e^4 - 5*b*c^4*e^5)*g)*x^4 + 2*(14*(7*c^5*d*e^4 - 2*b*c
^4*e^5)*f + (257*c^5*d^2*e^3 - 192*b*c^4*d*e^4 + 40*b^2*c^3*e^5)*g)*x^3 + 6*(7*(43*c^5*d^2*e^3 - 36*b*c^4*d*e^
4 + 8*b^2*c^3*e^5)*f + 2*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e^4 - 40*b^3*c^2*e^5)*g)*x^2 + 7
*(723*c^5*d^4*e - 1992*b*c^4*d^3*e^2 + 1968*b^2*c^3*d^2*e^3 - 832*b^3*c^2*d*e^4 + 128*b^4*c*e^5)*f + 2*(4707*c
^5*d^5 - 16563*b*c^4*d^4*e + 22752*b^2*c^3*d^3*e^2 - 15248*b^3*c^2*d^2*e^3 + 4992*b^4*c*d*e^4 - 640*b^5*e^5)*g
- 3*(28*(91*c^5*d^3*e^2 - 158*b*c^4*d^2*e^3 + 88*b^2*c^3*d*e^4 - 16*b^3*c^2*e^5)*f + (4707*c^5*d^4*e - 11856*
b*c^4*d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(e*x + d)/(c^8*e^5*x^3 + c^8*d^3*e^2 - 2*b*c^7*d^2*e^3 + b^2*c^6*d*e^4 - (c^8*d*e^4 - 2*b*c
^7*e^5)*x^2 - (c^8*d^2*e^3 - b^2*c^6*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(13/2)*(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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x