3.2250 \(\int (d+e x)^{5/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=501 \[ -\frac{512 (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{969 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2} \]

[Out]

(-512*(2*c*d - b*e)^5*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2909907*c^
7*e^2*(d + e*x)^(7/2)) - (256*(2*c*d - b*e)^4*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(7/2))/(415701*c^6*e^2*(d + e*x)^(5/2)) - (64*(2*c*d - b*e)^3*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(46189*c^5*e^2*(d + e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(19*c*e*f + 5*c*d*g -
12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(12597*c^4*e^2*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(19*c
*e*f + 5*c*d*g - 12*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(969*c^3*e^2) - (2*(19*c
*e*f + 5*c*d*g - 12*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(323*c^2*e^2) - (2*g*(
d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(19*c*e^2)

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Rubi [A]  time = 0.960119, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{512 (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{969 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-512*(2*c*d - b*e)^5*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2909907*c^
7*e^2*(d + e*x)^(7/2)) - (256*(2*c*d - b*e)^4*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(7/2))/(415701*c^6*e^2*(d + e*x)^(5/2)) - (64*(2*c*d - b*e)^3*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(46189*c^5*e^2*(d + e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(19*c*e*f + 5*c*d*g -
12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(12597*c^4*e^2*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(19*c
*e*f + 5*c*d*g - 12*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(969*c^3*e^2) - (2*(19*c
*e*f + 5*c*d*g - 12*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(323*c^2*e^2) - (2*g*(
d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(19*c*e^2)

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 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 (d+e x)^{5/2} (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 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{19 c e^3}\\ &=-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}+\frac{(10 (2 c d-b e) (19 c e f+5 c d g-12 b e g)) \int (d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{323 c^2 e}\\ &=-\frac{4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}+\frac{\left (16 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g)\right ) \int \sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{969 c^3 e}\\ &=-\frac{32 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}+\frac{\left (32 (2 c d-b e)^3 (19 c e f+5 c d g-12 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx}{4199 c^4 e}\\ &=-\frac{64 (2 c d-b e)^3 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}+\frac{\left (128 (2 c d-b e)^4 (19 c e f+5 c d g-12 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{46189 c^5 e}\\ &=-\frac{256 (2 c d-b e)^4 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}+\frac{\left (256 (2 c d-b e)^5 (19 c e f+5 c d g-12 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{415701 c^6 e}\\ &=-\frac{512 (2 c d-b e)^5 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac{256 (2 c d-b e)^4 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac{2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.780173, size = 284, normalized size = 0.57 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{7/2} \left (-171171 (b e-c d+c e x)^5 (-6 b e g+11 c d g+c e f)-969969 (2 c d-b e) (b e-c d+c e x)^4 (-3 b e g+5 c d g+c e f)-1322685 (2 c d-b e)^3 (b e-c d+c e x)^2 (-3 b e g+4 c d g+2 c e f)+2238390 (b e-2 c d)^2 (c (d-e x)-b e)^3 (-2 b e g+3 c d g+c e f)+323323 (b e-2 c d)^4 (c (d-e x)-b e) (-6 b e g+7 c d g+5 c e f)-415701 (2 c d-b e)^5 (-b e g+c d g+c e f)-153153 g (b e-c d+c e x)^6\right )}{2909907 c^7 e^2 (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(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)*(-(b*e) + c*(d - e*x)))^(7/2)*(-415701*(2*c*d - b*e)^5*(c*e*f + c*d*g - b*e*g) - 1322685*(2*c*d
- b*e)^3*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(-(c*d) + b*e + c*e*x)^2 - 969969*(2*c*d - b*e)*(c*e*f + 5*c*d*g - 3*b*
e*g)*(-(c*d) + b*e + c*e*x)^4 - 171171*(c*e*f + 11*c*d*g - 6*b*e*g)*(-(c*d) + b*e + c*e*x)^5 - 153153*g*(-(c*d
) + b*e + c*e*x)^6 + 323323*(-2*c*d + b*e)^4*(5*c*e*f + 7*c*d*g - 6*b*e*g)*(-(b*e) + c*(d - e*x)) + 2238390*(-
2*c*d + b*e)^2*(c*e*f + 3*c*d*g - 2*b*e*g)*(-(b*e) + c*(d - e*x))^3))/(2909907*c^7*e^2*(d + e*x)^(7/2))

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Maple [A]  time = 0.009, size = 739, normalized size = 1.5 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 153153\,g{e}^{6}{x}^{6}{c}^{6}-108108\,b{c}^{5}{e}^{6}g{x}^{5}+963963\,{c}^{6}d{e}^{5}g{x}^{5}+171171\,{c}^{6}{e}^{6}f{x}^{5}+72072\,{b}^{2}{c}^{4}{e}^{6}g{x}^{4}-714714\,b{c}^{5}d{e}^{5}g{x}^{4}-114114\,b{c}^{5}{e}^{6}f{x}^{4}+2582580\,{c}^{6}{d}^{2}{e}^{4}g{x}^{4}+1084083\,{c}^{6}d{e}^{5}f{x}^{4}-44352\,{b}^{3}{c}^{3}{e}^{6}g{x}^{3}+484176\,{b}^{2}{c}^{4}d{e}^{5}g{x}^{3}+70224\,{b}^{2}{c}^{4}{e}^{6}f{x}^{3}-2029104\,b{c}^{5}{d}^{2}{e}^{4}g{x}^{3}-737352\,b{c}^{5}d{e}^{5}f{x}^{3}+3827670\,{c}^{6}{d}^{3}{e}^{3}g{x}^{3}+2905518\,{c}^{6}{d}^{2}{e}^{4}f{x}^{3}+24192\,{b}^{4}{c}^{2}{e}^{6}g{x}^{2}-288288\,{b}^{3}{c}^{3}d{e}^{5}g{x}^{2}-38304\,{b}^{3}{c}^{3}{e}^{6}f{x}^{2}+1370880\,{b}^{2}{c}^{4}{d}^{2}{e}^{4}g{x}^{2}+440496\,{b}^{2}{c}^{4}d{e}^{5}f{x}^{2}-3194604\,b{c}^{5}{d}^{3}{e}^{3}g{x}^{2}-1987020\,b{c}^{5}{d}^{2}{e}^{4}f{x}^{2}+3410505\,{c}^{6}{d}^{4}{e}^{2}g{x}^{2}+4230198\,{c}^{6}{d}^{3}{e}^{3}f{x}^{2}-10752\,{b}^{5}c{e}^{6}gx+138880\,{b}^{4}{c}^{2}d{e}^{5}gx+17024\,{b}^{4}{c}^{2}{e}^{6}fx-737408\,{b}^{3}{c}^{3}{d}^{2}{e}^{4}gx-212800\,{b}^{3}{c}^{3}d{e}^{5}fx+2029104\,{b}^{2}{c}^{4}{d}^{3}{e}^{3}gx+1078896\,{b}^{2}{c}^{4}{d}^{2}{e}^{4}fx-2935604\,b{c}^{5}{d}^{4}{e}^{2}gx-2763208\,b{c}^{5}{d}^{3}{e}^{3}fx+1839103\,{c}^{6}{d}^{5}egx+3496703\,{c}^{6}{d}^{4}{e}^{2}fx+3072\,{b}^{6}{e}^{6}g-42752\,{b}^{5}cd{e}^{5}g-4864\,{b}^{5}c{e}^{6}f+250368\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}g+65664\,{b}^{4}{c}^{2}d{e}^{5}f-790432\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}g-369056\,{b}^{3}{c}^{3}{d}^{2}{e}^{4}f+1418488\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}g+1097744\,{b}^{2}{c}^{4}{d}^{3}{e}^{3}f-1364202\,b{c}^{5}{d}^{5}eg-1788546\,b{c}^{5}{d}^{4}{e}^{2}f+525458\,{c}^{6}{d}^{6}g+1414759\,f{d}^{5}{c}^{6}e \right ) }{2909907\,{c}^{7}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2909907*(c*e*x+b*e-c*d)*(153153*c^6*e^6*g*x^6-108108*b*c^5*e^6*g*x^5+963963*c^6*d*e^5*g*x^5+171171*c^6*e^6*f
*x^5+72072*b^2*c^4*e^6*g*x^4-714714*b*c^5*d*e^5*g*x^4-114114*b*c^5*e^6*f*x^4+2582580*c^6*d^2*e^4*g*x^4+1084083
*c^6*d*e^5*f*x^4-44352*b^3*c^3*e^6*g*x^3+484176*b^2*c^4*d*e^5*g*x^3+70224*b^2*c^4*e^6*f*x^3-2029104*b*c^5*d^2*
e^4*g*x^3-737352*b*c^5*d*e^5*f*x^3+3827670*c^6*d^3*e^3*g*x^3+2905518*c^6*d^2*e^4*f*x^3+24192*b^4*c^2*e^6*g*x^2
-288288*b^3*c^3*d*e^5*g*x^2-38304*b^3*c^3*e^6*f*x^2+1370880*b^2*c^4*d^2*e^4*g*x^2+440496*b^2*c^4*d*e^5*f*x^2-3
194604*b*c^5*d^3*e^3*g*x^2-1987020*b*c^5*d^2*e^4*f*x^2+3410505*c^6*d^4*e^2*g*x^2+4230198*c^6*d^3*e^3*f*x^2-107
52*b^5*c*e^6*g*x+138880*b^4*c^2*d*e^5*g*x+17024*b^4*c^2*e^6*f*x-737408*b^3*c^3*d^2*e^4*g*x-212800*b^3*c^3*d*e^
5*f*x+2029104*b^2*c^4*d^3*e^3*g*x+1078896*b^2*c^4*d^2*e^4*f*x-2935604*b*c^5*d^4*e^2*g*x-2763208*b*c^5*d^3*e^3*
f*x+1839103*c^6*d^5*e*g*x+3496703*c^6*d^4*e^2*f*x+3072*b^6*e^6*g-42752*b^5*c*d*e^5*g-4864*b^5*c*e^6*f+250368*b
^4*c^2*d^2*e^4*g+65664*b^4*c^2*d*e^5*f-790432*b^3*c^3*d^3*e^3*g-369056*b^3*c^3*d^2*e^4*f+1418488*b^2*c^4*d^4*e
^2*g+1097744*b^2*c^4*d^3*e^3*f-1364202*b*c^5*d^5*e*g-1788546*b*c^5*d^4*e^2*f+525458*c^6*d^6*g+1414759*c^6*d^5*
e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^7/e^2/(e*x+d)^(5/2)

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Maxima [B]  time = 1.43493, size = 1841, normalized size = 3.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*c^8*e^8*x^8 - 74461*c^8*d^8 + 317517*b*c^7*d^7*e - 563561*b^2*c^6*d^6*e^2 + 549615*b^3*c^5*d^5*
e^3 - 329190*b^4*c^4*d^4*e^4 + 126672*b^5*c^3*d^3*e^5 - 30560*b^6*c^2*d^2*e^6 + 4224*b^7*c*d*e^7 - 256*b^8*e^8
+ 3003*(10*c^8*d*e^7 + 7*b*c^7*e^8)*x^7 + 231*(38*c^8*d^2*e^6 + 417*b*c^7*d*e^7 + 55*b^2*c^6*e^8)*x^6 - 63*(1
174*c^8*d^3*e^5 - 2179*b*c^7*d^2*e^6 - 1204*b^2*c^6*d*e^7 - b^3*c^5*e^8)*x^5 - 35*(2348*c^8*d^4*e^4 + 587*b*c^
7*d^3*e^5 - 5343*b^2*c^6*d^2*e^6 - 25*b^3*c^5*d*e^7 + 2*b^4*c^4*e^8)*x^4 + (37354*c^8*d^5*e^3 - 257745*b*c^7*d
^4*e^4 + 237200*b^2*c^6*d^3*e^5 + 6070*b^3*c^5*d^2*e^6 - 1080*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*x^3 + 3*(35362*c
^8*d^6*e^2 - 87409*b*c^7*d^5*e^3 + 44825*b^2*c^6*d^4*e^4 + 9650*b^3*c^5*d^3*e^5 - 2860*b^4*c^4*d^2*e^6 + 464*b
^5*c^3*d*e^7 - 32*b^6*c^2*e^8)*x^2 + (39346*c^8*d^7*e - 31625*b*c^7*d^6*e^2 - 83676*b^2*c^6*d^5*e^3 + 114555*b
^3*c^5*d^4*e^4 - 50040*b^4*c^4*d^3*e^5 + 13296*b^5*c^3*d^2*e^6 - 1984*b^6*c^2*d*e^7 + 128*b^7*c*e^8)*x)*sqrt(-
c*e*x + c*d - b*e)*(e*x + d)*f/(c^6*e^2*x + c^6*d*e) + 2/2909907*(153153*c^9*e^9*x^9 - 525458*c^9*d^9 + 294057
6*b*c^8*d^8*e - 7087468*b^2*c^7*d^7*e^2 + 9663960*b^3*c^6*d^6*e^3 - 8241330*b^4*c^5*d^5*e^4 + 4583640*b^5*c^4*
d^4*e^5 - 1672864*b^6*c^3*d^3*e^6 + 387840*b^7*c^2*d^2*e^7 - 51968*b^8*c*d*e^8 + 3072*b^9*e^9 + 9009*(56*c^9*d
*e^8 + 39*b*c^8*e^9)*x^8 + 3003*(50*c^9*d^2*e^7 + 527*b*c^8*d*e^8 + 69*b^2*c^7*e^9)*x^7 - 231*(5114*c^9*d^3*e^
6 - 9585*b*c^8*d^2*e^7 - 5216*b^2*c^7*d*e^8 - 3*b^3*c^6*e^9)*x^6 - 63*(20456*c^9*d^4*e^5 + 4189*b*c^8*d^3*e^6
- 45509*b^2*c^7*d^2*e^7 - 143*b^3*c^6*d*e^8 + 12*b^4*c^5*e^9)*x^5 + 7*(72574*c^9*d^5*e^4 - 530165*b*c^8*d^4*e^
5 + 496980*b^2*c^7*d^3*e^6 + 8230*b^3*c^6*d^2*e^7 - 1550*b^4*c^5*d*e^8 + 120*b^5*c^4*e^9)*x^4 + (1411994*c^9*d
^6*e^3 - 3574809*b*c^8*d^5*e^4 + 1981645*b^2*c^7*d^4*e^5 + 247010*b^3*c^6*d^3*e^6 - 78240*b^4*c^5*d^2*e^7 + 13
360*b^5*c^4*d*e^8 - 960*b^6*c^3*e^9)*x^3 + 3*(176810*c^9*d^7*e^2 - 248777*b*c^8*d^6*e^3 - 105344*b^2*c^7*d^5*e
^4 + 276115*b^3*c^6*d^4*e^5 - 130100*b^4*c^5*d^3*e^6 + 36640*b^5*c^4*d^2*e^7 - 5728*b^6*c^3*d*e^8 + 384*b^7*c^
2*e^9)*x^2 - (262729*c^9*d^8*e - 1207559*b*c^8*d^7*e^2 + 2336175*b^2*c^7*d^6*e^3 - 2495805*b^3*c^6*d^5*e^4 + 1
624860*b^4*c^5*d^4*e^5 - 666960*b^5*c^4*d^3*e^6 + 169472*b^6*c^3*d^2*e^7 - 24448*b^7*c^2*d*e^8 + 1536*b^8*c*e^
9)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^7*e^3*x + c^7*d*e^2)

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Fricas [B]  time = 1.84613, size = 3205, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2909907*(153153*c^9*e^9*g*x^9 + 9009*(19*c^9*e^9*f + (56*c^9*d*e^8 + 39*b*c^8*e^9)*g)*x^8 + 3003*(19*(10*c^9
*d*e^8 + 7*b*c^8*e^9)*f + (50*c^9*d^2*e^7 + 527*b*c^8*d*e^8 + 69*b^2*c^7*e^9)*g)*x^7 + 231*(19*(38*c^9*d^2*e^7
+ 417*b*c^8*d*e^8 + 55*b^2*c^7*e^9)*f - (5114*c^9*d^3*e^6 - 9585*b*c^8*d^2*e^7 - 5216*b^2*c^7*d*e^8 - 3*b^3*c
^6*e^9)*g)*x^6 - 63*(19*(1174*c^9*d^3*e^6 - 2179*b*c^8*d^2*e^7 - 1204*b^2*c^7*d*e^8 - b^3*c^6*e^9)*f + (20456*
c^9*d^4*e^5 + 4189*b*c^8*d^3*e^6 - 45509*b^2*c^7*d^2*e^7 - 143*b^3*c^6*d*e^8 + 12*b^4*c^5*e^9)*g)*x^5 - 7*(95*
(2348*c^9*d^4*e^5 + 587*b*c^8*d^3*e^6 - 5343*b^2*c^7*d^2*e^7 - 25*b^3*c^6*d*e^8 + 2*b^4*c^5*e^9)*f - (72574*c^
9*d^5*e^4 - 530165*b*c^8*d^4*e^5 + 496980*b^2*c^7*d^3*e^6 + 8230*b^3*c^6*d^2*e^7 - 1550*b^4*c^5*d*e^8 + 120*b^
5*c^4*e^9)*g)*x^4 + (19*(37354*c^9*d^5*e^4 - 257745*b*c^8*d^4*e^5 + 237200*b^2*c^7*d^3*e^6 + 6070*b^3*c^6*d^2*
e^7 - 1080*b^4*c^5*d*e^8 + 80*b^5*c^4*e^9)*f + (1411994*c^9*d^6*e^3 - 3574809*b*c^8*d^5*e^4 + 1981645*b^2*c^7*
d^4*e^5 + 247010*b^3*c^6*d^3*e^6 - 78240*b^4*c^5*d^2*e^7 + 13360*b^5*c^4*d*e^8 - 960*b^6*c^3*e^9)*g)*x^3 + 3*(
19*(35362*c^9*d^6*e^3 - 87409*b*c^8*d^5*e^4 + 44825*b^2*c^7*d^4*e^5 + 9650*b^3*c^6*d^3*e^6 - 2860*b^4*c^5*d^2*
e^7 + 464*b^5*c^4*d*e^8 - 32*b^6*c^3*e^9)*f + (176810*c^9*d^7*e^2 - 248777*b*c^8*d^6*e^3 - 105344*b^2*c^7*d^5*
e^4 + 276115*b^3*c^6*d^4*e^5 - 130100*b^4*c^5*d^3*e^6 + 36640*b^5*c^4*d^2*e^7 - 5728*b^6*c^3*d*e^8 + 384*b^7*c
^2*e^9)*g)*x^2 - 19*(74461*c^9*d^8*e - 317517*b*c^8*d^7*e^2 + 563561*b^2*c^7*d^6*e^3 - 549615*b^3*c^6*d^5*e^4
+ 329190*b^4*c^5*d^4*e^5 - 126672*b^5*c^4*d^3*e^6 + 30560*b^6*c^3*d^2*e^7 - 4224*b^7*c^2*d*e^8 + 256*b^8*c*e^9
)*f - 2*(262729*c^9*d^9 - 1470288*b*c^8*d^8*e + 3543734*b^2*c^7*d^7*e^2 - 4831980*b^3*c^6*d^6*e^3 + 4120665*b^
4*c^5*d^5*e^4 - 2291820*b^5*c^4*d^4*e^5 + 836432*b^6*c^3*d^3*e^6 - 193920*b^7*c^2*d^2*e^7 + 25984*b^8*c*d*e^8
- 1536*b^9*e^9)*g + (19*(39346*c^9*d^7*e^2 - 31625*b*c^8*d^6*e^3 - 83676*b^2*c^7*d^5*e^4 + 114555*b^3*c^6*d^4*
e^5 - 50040*b^4*c^5*d^3*e^6 + 13296*b^5*c^4*d^2*e^7 - 1984*b^6*c^3*d*e^8 + 128*b^7*c^2*e^9)*f - (262729*c^9*d^
8*e - 1207559*b*c^8*d^7*e^2 + 2336175*b^2*c^7*d^6*e^3 - 2495805*b^3*c^6*d^5*e^4 + 1624860*b^4*c^5*d^4*e^5 - 66
6960*b^5*c^4*d^3*e^6 + 169472*b^6*c^3*d^2*e^7 - 24448*b^7*c^2*d*e^8 + 1536*b^8*c*e^9)*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^3*x + c^7*d*e^2)

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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)**(5/2)*(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(-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)^(5/2)*(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