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

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

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Rubi [A]  time = 0.96, antiderivative size = 501, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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 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])

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*}

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Mathematica [A]  time = 0.70, size = 284, normalized size = 0.57 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 4.96, size = 839, normalized size = 1.67 \begin {gather*} -\frac {2 \left ((2 c d-b e) (d+e x)-c (d+e x)^2\right )^{7/2} \left (153153 g (d+e x)^6 c^6+171171 e f (d+e x)^5 c^6+45045 d g (d+e x)^5 c^6+228228 d e f (d+e x)^4 c^6+60060 d^2 g (d+e x)^4 c^6+280896 d^2 e f (d+e x)^3 c^6+73920 d^3 g (d+e x)^3 c^6+306432 d^3 e f (d+e x)^2 c^6+80640 d^4 g (d+e x)^2 c^6+155648 d^5 e f c^6+40960 d^6 g c^6+272384 d^4 e f (d+e x) c^6+71680 d^5 g (d+e x) c^6-108108 b e g (d+e x)^5 c^5-114114 b e^2 f (d+e x)^4 c^5-174174 b d e g (d+e x)^4 c^5-280896 b d e^2 f (d+e x)^3 c^5-251328 b d^2 e g (d+e x)^3 c^5-459648 b d^2 e^2 f (d+e x)^2 c^5-314496 b d^3 e g (d+e x)^2 c^5-389120 b d^4 e^2 f c^5-200704 b d^5 e g c^5-544768 b d^3 e^2 f (d+e x) c^5-315392 b d^4 e g (d+e x) c^5+72072 b^2 e^2 g (d+e x)^4 c^4+70224 b^2 e^3 f (d+e x)^3 c^4+195888 b^2 d e^2 g (d+e x)^3 c^4+229824 b^2 d e^3 f (d+e x)^2 c^4+350784 b^2 d^2 e^2 g (d+e x)^2 c^4+389120 b^2 d^3 e^3 f c^4+348160 b^2 d^4 e^2 g c^4+408576 b^2 d^2 e^3 f (d+e x) c^4+451584 b^2 d^3 e^2 g (d+e x) c^4-44352 b^3 e^3 g (d+e x)^3 c^3-38304 b^3 e^4 f (d+e x)^2 c^3-155232 b^3 d e^3 g (d+e x)^2 c^3-194560 b^3 d^2 e^4 f c^3-296960 b^3 d^3 e^3 g c^3-136192 b^3 d e^4 f (d+e x) c^3-293888 b^3 d^2 e^3 g (d+e x) c^3+24192 b^4 e^4 g (d+e x)^2 c^2+48640 b^4 d e^5 f c^2+135680 b^4 d^2 e^4 g c^2+17024 b^4 e^5 f (d+e x) c^2+90496 b^4 d e^4 g (d+e x) c^2-4864 b^5 e^6 f c-32000 b^5 d e^5 g c-10752 b^5 e^5 g (d+e x) c+3072 b^6 e^6 g\right )}{2909907 c^7 e^2 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(7/2)*(155648*c^6*d^5*e*f - 389120*b*c^5*d^4*e^2*f + 389120*b^2*
c^4*d^3*e^3*f - 194560*b^3*c^3*d^2*e^4*f + 48640*b^4*c^2*d*e^5*f - 4864*b^5*c*e^6*f + 40960*c^6*d^6*g - 200704
*b*c^5*d^5*e*g + 348160*b^2*c^4*d^4*e^2*g - 296960*b^3*c^3*d^3*e^3*g + 135680*b^4*c^2*d^2*e^4*g - 32000*b^5*c*
d*e^5*g + 3072*b^6*e^6*g + 272384*c^6*d^4*e*f*(d + e*x) - 544768*b*c^5*d^3*e^2*f*(d + e*x) + 408576*b^2*c^4*d^
2*e^3*f*(d + e*x) - 136192*b^3*c^3*d*e^4*f*(d + e*x) + 17024*b^4*c^2*e^5*f*(d + e*x) + 71680*c^6*d^5*g*(d + e*
x) - 315392*b*c^5*d^4*e*g*(d + e*x) + 451584*b^2*c^4*d^3*e^2*g*(d + e*x) - 293888*b^3*c^3*d^2*e^3*g*(d + e*x)
+ 90496*b^4*c^2*d*e^4*g*(d + e*x) - 10752*b^5*c*e^5*g*(d + e*x) + 306432*c^6*d^3*e*f*(d + e*x)^2 - 459648*b*c^
5*d^2*e^2*f*(d + e*x)^2 + 229824*b^2*c^4*d*e^3*f*(d + e*x)^2 - 38304*b^3*c^3*e^4*f*(d + e*x)^2 + 80640*c^6*d^4
*g*(d + e*x)^2 - 314496*b*c^5*d^3*e*g*(d + e*x)^2 + 350784*b^2*c^4*d^2*e^2*g*(d + e*x)^2 - 155232*b^3*c^3*d*e^
3*g*(d + e*x)^2 + 24192*b^4*c^2*e^4*g*(d + e*x)^2 + 280896*c^6*d^2*e*f*(d + e*x)^3 - 280896*b*c^5*d*e^2*f*(d +
e*x)^3 + 70224*b^2*c^4*e^3*f*(d + e*x)^3 + 73920*c^6*d^3*g*(d + e*x)^3 - 251328*b*c^5*d^2*e*g*(d + e*x)^3 + 1
95888*b^2*c^4*d*e^2*g*(d + e*x)^3 - 44352*b^3*c^3*e^3*g*(d + e*x)^3 + 228228*c^6*d*e*f*(d + e*x)^4 - 114114*b*
c^5*e^2*f*(d + e*x)^4 + 60060*c^6*d^2*g*(d + e*x)^4 - 174174*b*c^5*d*e*g*(d + e*x)^4 + 72072*b^2*c^4*e^2*g*(d
+ e*x)^4 + 171171*c^6*e*f*(d + e*x)^5 + 45045*c^6*d*g*(d + e*x)^5 - 108108*b*c^5*e*g*(d + e*x)^5 + 153153*c^6*
g*(d + e*x)^6))/(2909907*c^7*e^2*(d + e*x)^(7/2))

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fricas [B]  time = 0.50, size = 1370, normalized size = 2.73

result too large to display

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

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]

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

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maple [A]  time = 0.05, size = 739, normalized size = 1.48 \begin {gather*} \frac {2 \left (c e x +b e -c d \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} 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 f \,d^{5} c^{6} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{2909907 \left (e x +d \right )^{\frac {5}{2}} c^{7} e^{2}} \end {gather*}

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 = 1.03, size = 1364, normalized size = 2.72

result too large to display

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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mupad [B]  time = 5.39, size = 1307, normalized size = 2.61 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^4\,x^7\,\sqrt {d+e\,x}\,\left (69\,g\,b^2\,e^2+527\,g\,b\,c\,d\,e+133\,f\,b\,c\,e^2+50\,g\,c^2\,d^2+190\,f\,c^2\,d\,e\right )}{969}+\frac {x^5\,\sqrt {d+e\,x}\,\left (-1512\,g\,b^4\,c^5\,e^9+18018\,g\,b^3\,c^6\,d\,e^8+2394\,f\,b^3\,c^6\,e^9+5734134\,g\,b^2\,c^7\,d^2\,e^7+2882376\,f\,b^2\,c^7\,d\,e^8-527814\,g\,b\,c^8\,d^3\,e^6+5216526\,f\,b\,c^8\,d^2\,e^7-2577456\,g\,c^9\,d^4\,e^5-2810556\,f\,c^9\,d^3\,e^6\right )}{2909907\,c^7\,e^3}+\frac {2\,c^2\,e^6\,g\,x^9\,\sqrt {d+e\,x}}{19}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-1920\,g\,b^6\,c^3\,e^9+26720\,g\,b^5\,c^4\,d\,e^8+3040\,f\,b^5\,c^4\,e^9-156480\,g\,b^4\,c^5\,d^2\,e^7-41040\,f\,b^4\,c^5\,d\,e^8+494020\,g\,b^3\,c^6\,d^3\,e^6+230660\,f\,b^3\,c^6\,d^2\,e^7+3963290\,g\,b^2\,c^7\,d^4\,e^5+9013600\,f\,b^2\,c^7\,d^3\,e^6-7149618\,g\,b\,c^8\,d^5\,e^4-9794310\,f\,b\,c^8\,d^4\,e^5+2823988\,g\,c^9\,d^6\,e^3+1419452\,f\,c^9\,d^5\,e^4\right )}{2909907\,c^7\,e^3}+\frac {x^6\,\sqrt {d+e\,x}\,\left (1386\,g\,b^3\,c^6\,e^9+2409792\,g\,b^2\,c^7\,d\,e^8+482790\,f\,b^2\,c^7\,e^9+4428270\,g\,b\,c^8\,d^2\,e^7+3660426\,f\,b\,c^8\,d\,e^8-2362668\,g\,c^9\,d^3\,e^6+333564\,f\,c^9\,d^2\,e^7\right )}{2909907\,c^7\,e^3}+\frac {2\,c\,e^5\,x^8\,\sqrt {d+e\,x}\,\left (39\,b\,e\,g+56\,c\,d\,g+19\,c\,e\,f\right )}{323}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (3072\,g\,b^6\,e^6-42752\,g\,b^5\,c\,d\,e^5-4864\,f\,b^5\,c\,e^6+250368\,g\,b^4\,c^2\,d^2\,e^4+65664\,f\,b^4\,c^2\,d\,e^5-790432\,g\,b^3\,c^3\,d^3\,e^3-369056\,f\,b^3\,c^3\,d^2\,e^4+1418488\,g\,b^2\,c^4\,d^4\,e^2+1097744\,f\,b^2\,c^4\,d^3\,e^3-1364202\,g\,b\,c^5\,d^5\,e-1788546\,f\,b\,c^5\,d^4\,e^2+525458\,g\,c^6\,d^6+1414759\,f\,c^6\,d^5\,e\right )}{2909907\,c^7\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (1680\,g\,b^5\,c^4\,e^9-21700\,g\,b^4\,c^5\,d\,e^8-2660\,f\,b^4\,c^5\,e^9+115220\,g\,b^3\,c^6\,d^2\,e^7+33250\,f\,b^3\,c^6\,d\,e^8+6957720\,g\,b^2\,c^7\,d^3\,e^6+7106190\,f\,b^2\,c^7\,d^2\,e^7-7422310\,g\,b\,c^8\,d^4\,e^5-780710\,f\,b\,c^8\,d^3\,e^6+1016036\,g\,c^9\,d^5\,e^4-3122840\,f\,c^9\,d^4\,e^5\right )}{2909907\,c^7\,e^3}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (384\,g\,b^6\,e^6-5344\,g\,b^5\,c\,d\,e^5-608\,f\,b^5\,c\,e^6+31296\,g\,b^4\,c^2\,d^2\,e^4+8208\,f\,b^4\,c^2\,d\,e^5-98804\,g\,b^3\,c^3\,d^3\,e^3-46132\,f\,b^3\,c^3\,d^2\,e^4+177311\,g\,b^2\,c^4\,d^4\,e^2+137218\,f\,b^2\,c^4\,d^3\,e^3+71967\,g\,b\,c^5\,d^5\,e+988893\,f\,b\,c^5\,d^4\,e^2-176810\,g\,c^6\,d^6-671878\,f\,c^6\,d^5\,e\right )}{969969\,c^5\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-1536\,g\,b^6\,e^6+21376\,g\,b^5\,c\,d\,e^5+2432\,f\,b^5\,c\,e^6-125184\,g\,b^4\,c^2\,d^2\,e^4-32832\,f\,b^4\,c^2\,d\,e^5+395216\,g\,b^3\,c^3\,d^3\,e^3+184528\,f\,b^3\,c^3\,d^2\,e^4-709244\,g\,b^2\,c^4\,d^4\,e^2-548872\,f\,b^2\,c^4\,d^3\,e^3+682101\,g\,b\,c^5\,d^5\,e+894273\,f\,b\,c^5\,d^4\,e^2-262729\,g\,c^6\,d^6+747574\,f\,c^6\,d^5\,e\right )}{2909907\,c^6\,e^2}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)*(d + e*x)^(5/2)*(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)*((2*e^4*x^7*(d + e*x)^(1/2)*(69*b^2*e^2*g + 50*c^2*d^2*g + 133*b*
c*e^2*f + 190*c^2*d*e*f + 527*b*c*d*e*g))/969 + (x^5*(d + e*x)^(1/2)*(2394*b^3*c^6*e^9*f - 1512*b^4*c^5*e^9*g
- 2810556*c^9*d^3*e^6*f - 2577456*c^9*d^4*e^5*g + 5216526*b*c^8*d^2*e^7*f + 2882376*b^2*c^7*d*e^8*f - 527814*b
*c^8*d^3*e^6*g + 18018*b^3*c^6*d*e^8*g + 5734134*b^2*c^7*d^2*e^7*g))/(2909907*c^7*e^3) + (2*c^2*e^6*g*x^9*(d +
e*x)^(1/2))/19 + (x^3*(d + e*x)^(1/2)*(3040*b^5*c^4*e^9*f - 1920*b^6*c^3*e^9*g + 1419452*c^9*d^5*e^4*f + 2823
988*c^9*d^6*e^3*g - 9794310*b*c^8*d^4*e^5*f - 41040*b^4*c^5*d*e^8*f - 7149618*b*c^8*d^5*e^4*g + 26720*b^5*c^4*
d*e^8*g + 9013600*b^2*c^7*d^3*e^6*f + 230660*b^3*c^6*d^2*e^7*f + 3963290*b^2*c^7*d^4*e^5*g + 494020*b^3*c^6*d^
3*e^6*g - 156480*b^4*c^5*d^2*e^7*g))/(2909907*c^7*e^3) + (x^6*(d + e*x)^(1/2)*(482790*b^2*c^7*e^9*f + 1386*b^3
*c^6*e^9*g + 333564*c^9*d^2*e^7*f - 2362668*c^9*d^3*e^6*g + 3660426*b*c^8*d*e^8*f + 4428270*b*c^8*d^2*e^7*g +
2409792*b^2*c^7*d*e^8*g))/(2909907*c^7*e^3) + (2*c*e^5*x^8*(d + e*x)^(1/2)*(39*b*e*g + 56*c*d*g + 19*c*e*f))/3
23 + (2*(b*e - c*d)^3*(d + e*x)^(1/2)*(3072*b^6*e^6*g + 525458*c^6*d^6*g - 4864*b^5*c*e^6*f + 1414759*c^6*d^5*
e*f - 1364202*b*c^5*d^5*e*g - 42752*b^5*c*d*e^5*g - 1788546*b*c^5*d^4*e^2*f + 65664*b^4*c^2*d*e^5*f + 1097744*
b^2*c^4*d^3*e^3*f - 369056*b^3*c^3*d^2*e^4*f + 1418488*b^2*c^4*d^4*e^2*g - 790432*b^3*c^3*d^3*e^3*g + 250368*b
^4*c^2*d^2*e^4*g))/(2909907*c^7*e^3) + (x^4*(d + e*x)^(1/2)*(1680*b^5*c^4*e^9*g - 2660*b^4*c^5*e^9*f - 3122840
*c^9*d^4*e^5*f + 1016036*c^9*d^5*e^4*g - 780710*b*c^8*d^3*e^6*f + 33250*b^3*c^6*d*e^8*f - 7422310*b*c^8*d^4*e^
5*g - 21700*b^4*c^5*d*e^8*g + 7106190*b^2*c^7*d^2*e^7*f + 6957720*b^2*c^7*d^3*e^6*g + 115220*b^3*c^6*d^2*e^7*g
))/(2909907*c^7*e^3) + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(384*b^6*e^6*g - 176810*c^6*d^6*g - 608*b^5*c*e^6*f
- 671878*c^6*d^5*e*f + 71967*b*c^5*d^5*e*g - 5344*b^5*c*d*e^5*g + 988893*b*c^5*d^4*e^2*f + 8208*b^4*c^2*d*e^5*
f + 137218*b^2*c^4*d^3*e^3*f - 46132*b^3*c^3*d^2*e^4*f + 177311*b^2*c^4*d^4*e^2*g - 98804*b^3*c^3*d^3*e^3*g +
31296*b^4*c^2*d^2*e^4*g))/(969969*c^5*e) + (2*x*(b*e - c*d)^2*(d + e*x)^(1/2)*(2432*b^5*c*e^6*f - 262729*c^6*d
^6*g - 1536*b^6*e^6*g + 747574*c^6*d^5*e*f + 682101*b*c^5*d^5*e*g + 21376*b^5*c*d*e^5*g + 894273*b*c^5*d^4*e^2
*f - 32832*b^4*c^2*d*e^5*f - 548872*b^2*c^4*d^3*e^3*f + 184528*b^3*c^3*d^2*e^4*f - 709244*b^2*c^4*d^4*e^2*g +
395216*b^3*c^3*d^3*e^3*g - 125184*b^4*c^2*d^2*e^4*g))/(2909907*c^6*e^2)))/(x + d/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)**(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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