3.21.11 \(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=347 \[ -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \]

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Rubi [A]  time = 0.60, antiderivative size = 347, 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 = {794, 656, 648} \begin {gather*} -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-32*(2*c*d - b*e)^3*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15015*c^5*e^
2*(d + e*x)^(5/2)) - (16*(2*c*d - b*e)^2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^
(5/2))/(3003*c^4*e^2*(d + e*x)^(3/2)) - (4*(2*c*d - b*e)*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2
*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2*Sqrt[d + e*x]) - (2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(143*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(13*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)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (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}{13 c e^3}\\ &=-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {(6 (2 c d-b e) (13 c e f+3 c d g-8 b e g)) \int \sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{143 c^2 e}\\ &=-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {\left (8 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{429 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {\left (16 (2 c d-b e)^3 (13 c e f+3 c d g-8 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 c^4 e}\\ &=-\frac {32 (2 c d-b e)^3 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 264, normalized size = 0.76 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (71 d g+13 e f+20 e g x)+8 b^2 c^2 e^2 \left (473 d^2 g+d e (221 f+315 g x)+5 e^2 x (13 f+14 g x)\right )-2 b c^3 e \left (2765 d^3 g+d^2 e (2743 f+3470 g x)+25 d e^2 x (78 f+77 g x)+35 e^3 x^2 (13 f+12 g x)\right )+c^4 \left (2754 d^4 g+d^3 e (6929 f+6885 g x)+5 d^2 e^2 x (1963 f+1659 g x)+35 d e^3 x^2 (169 f+141 g x)+105 e^4 x^3 (13 f+11 g x)\right )\right )}{15015 c^5 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4*e^4*g - 16*b^3*c*e^3*(13*e*f + 71
*d*g + 20*e*g*x) + 8*b^2*c^2*e^2*(473*d^2*g + 5*e^2*x*(13*f + 14*g*x) + d*e*(221*f + 315*g*x)) - 2*b*c^3*e*(27
65*d^3*g + 35*e^3*x^2*(13*f + 12*g*x) + 25*d*e^2*x*(78*f + 77*g*x) + d^2*e*(2743*f + 3470*g*x)) + c^4*(2754*d^
4*g + 105*e^4*x^3*(13*f + 11*g*x) + 35*d*e^3*x^2*(169*f + 141*g*x) + 5*d^2*e^2*x*(1963*f + 1659*g*x) + d^3*e*(
6929*f + 6885*g*x))))/(15015*c^5*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 3.97, size = 401, normalized size = 1.16 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{5/2} \left (128 b^4 e^4 g-320 b^3 c e^3 g (d+e x)-816 b^3 c d e^3 g-208 b^3 c e^4 f+1824 b^2 c^2 d^2 e^2 g+520 b^2 c^2 e^3 f (d+e x)+1248 b^2 c^2 d e^3 f+560 b^2 c^2 e^2 g (d+e x)^2+1400 b^2 c^2 d e^2 g (d+e x)-1600 b c^3 d^3 e g-2496 b c^3 d^2 e^2 f-1760 b c^3 d^2 e g (d+e x)-910 b c^3 e^2 f (d+e x)^2-2080 b c^3 d e^2 f (d+e x)-840 b c^3 e g (d+e x)^3-1330 b c^3 d e g (d+e x)^2+384 c^4 d^4 g+1664 c^4 d^3 e f+480 c^4 d^3 g (d+e x)+2080 c^4 d^2 e f (d+e x)+420 c^4 d^2 g (d+e x)^2+1365 c^4 e f (d+e x)^3+1820 c^4 d e f (d+e x)^2+1155 c^4 g (d+e x)^4+315 c^4 d g (d+e x)^3\right )}{15015 c^5 e^2 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(5/2)*(1664*c^4*d^3*e*f - 2496*b*c^3*d^2*e^2*f + 1248*b^2*c^2*d*
e^3*f - 208*b^3*c*e^4*f + 384*c^4*d^4*g - 1600*b*c^3*d^3*e*g + 1824*b^2*c^2*d^2*e^2*g - 816*b^3*c*d*e^3*g + 12
8*b^4*e^4*g + 2080*c^4*d^2*e*f*(d + e*x) - 2080*b*c^3*d*e^2*f*(d + e*x) + 520*b^2*c^2*e^3*f*(d + e*x) + 480*c^
4*d^3*g*(d + e*x) - 1760*b*c^3*d^2*e*g*(d + e*x) + 1400*b^2*c^2*d*e^2*g*(d + e*x) - 320*b^3*c*e^3*g*(d + e*x)
+ 1820*c^4*d*e*f*(d + e*x)^2 - 910*b*c^3*e^2*f*(d + e*x)^2 + 420*c^4*d^2*g*(d + e*x)^2 - 1330*b*c^3*d*e*g*(d +
e*x)^2 + 560*b^2*c^2*e^2*g*(d + e*x)^2 + 1365*c^4*e*f*(d + e*x)^3 + 315*c^4*d*g*(d + e*x)^3 - 840*b*c^3*e*g*(
d + e*x)^3 + 1155*c^4*g*(d + e*x)^4))/(15015*c^5*e^2*(d + e*x)^(5/2))

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fricas [B]  time = 0.42, size = 678, normalized size = 1.95 \begin {gather*} -\frac {2 \, {\left (1155 \, c^{6} e^{6} g x^{6} + 105 \, {\left (13 \, c^{6} e^{6} f + {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} g\right )} x^{5} + 35 \, {\left (13 \, {\left (7 \, c^{6} d e^{5} + 4 \, b c^{5} e^{6}\right )} f - {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - 5 \, {\left (13 \, {\left (10 \, c^{6} d^{2} e^{4} - 108 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} f + {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} - 3 \, {\left (13 \, {\left (174 \, c^{6} d^{3} e^{3} - 236 \, b c^{5} d^{2} e^{4} - 17 \, b^{2} c^{4} d e^{5} + 2 \, b^{3} c^{3} e^{6}\right )} f + {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} + 13 \, {\left (533 \, c^{6} d^{5} e - 1488 \, b c^{5} d^{4} e^{2} + 1513 \, b^{2} c^{4} d^{3} e^{3} - 710 \, b^{3} c^{3} d^{2} e^{4} + 168 \, b^{4} c^{2} d e^{5} - 16 \, b^{5} c e^{6}\right )} f + 2 \, {\left (1377 \, c^{6} d^{6} - 5519 \, b c^{5} d^{5} e + 8799 \, b^{2} c^{4} d^{4} e^{2} - 7117 \, b^{3} c^{3} d^{3} e^{3} + 3092 \, b^{4} c^{2} d^{2} e^{4} - 696 \, b^{5} c d e^{5} + 64 \, b^{6} e^{6}\right )} g - {\left (13 \, {\left (311 \, c^{6} d^{4} e^{2} - 100 \, b c^{5} d^{3} e^{3} - 279 \, b^{2} c^{4} d^{2} e^{4} + 76 \, b^{3} c^{3} d e^{5} - 8 \, b^{4} c^{2} e^{6}\right )} f - {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\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}}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/15015*(1155*c^6*e^6*g*x^6 + 105*(13*c^6*e^6*f + (25*c^6*d*e^5 + 14*b*c^5*e^6)*g)*x^5 + 35*(13*(7*c^6*d*e^5
+ 4*b*c^5*e^6)*f - (12*c^6*d^2*e^4 - 154*b*c^5*d*e^5 - b^2*c^4*e^6)*g)*x^4 - 5*(13*(10*c^6*d^2*e^4 - 108*b*c^5
*d*e^5 - b^2*c^4*e^6)*f + (954*c^6*d^3*e^3 - 1328*b*c^5*d^2*e^4 - 63*b^2*c^4*d*e^5 + 8*b^3*c^3*e^6)*g)*x^3 - 3
*(13*(174*c^6*d^3*e^3 - 236*b*c^5*d^2*e^4 - 17*b^2*c^4*d*e^5 + 2*b^3*c^3*e^6)*f + (907*c^6*d^4*e^2 - 560*b*c^5
*d^3*e^3 - 473*b^2*c^4*d^2*e^4 + 142*b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*g)*x^2 + 13*(533*c^6*d^5*e - 1488*b*c^5*d
^4*e^2 + 1513*b^2*c^4*d^3*e^3 - 710*b^3*c^3*d^2*e^4 + 168*b^4*c^2*d*e^5 - 16*b^5*c*e^6)*f + 2*(1377*c^6*d^6 -
5519*b*c^5*d^5*e + 8799*b^2*c^4*d^4*e^2 - 7117*b^3*c^3*d^3*e^3 + 3092*b^4*c^2*d^2*e^4 - 696*b^5*c*d*e^5 + 64*b
^6*e^6)*g - (13*(311*c^6*d^4*e^2 - 100*b*c^5*d^3*e^3 - 279*b^2*c^4*d^2*e^4 + 76*b^3*c^3*d*e^5 - 8*b^4*c^2*e^6)
*f - (1377*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657*b^2*c^4*d^3*e^3 - 2460*b^3*c^3*d^2*e^4 + 632*b^4*c^2*d*e^5 -
64*b^5*c*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^5*e^3*x + c^5*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 {3}{2}} {\left (e x + d\right )}^{\frac {3}{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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(3/2)*(g*x + f), x)

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maple [A]  time = 0.05, size = 367, normalized size = 1.06 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (1155 g \,e^{4} x^{4} c^{4}-840 b \,c^{3} e^{4} g \,x^{3}+4935 c^{4} d \,e^{3} g \,x^{3}+1365 c^{4} e^{4} f \,x^{3}+560 b^{2} c^{2} e^{4} g \,x^{2}-3850 b \,c^{3} d \,e^{3} g \,x^{2}-910 b \,c^{3} e^{4} f \,x^{2}+8295 c^{4} d^{2} e^{2} g \,x^{2}+5915 c^{4} d \,e^{3} f \,x^{2}-320 b^{3} c \,e^{4} g x +2520 b^{2} c^{2} d \,e^{3} g x +520 b^{2} c^{2} e^{4} f x -6940 b \,c^{3} d^{2} e^{2} g x -3900 b \,c^{3} d \,e^{3} f x +6885 c^{4} d^{3} e g x +9815 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1136 b^{3} c d \,e^{3} g -208 b^{3} c \,e^{4} f +3784 b^{2} c^{2} d^{2} e^{2} g +1768 b^{2} c^{2} d \,e^{3} f -5530 b \,c^{3} d^{3} e g -5486 b \,c^{3} d^{2} e^{2} f +2754 c^{4} d^{4} g +6929 f \,d^{3} c^{4} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {3}{2}} c^{5} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

2/15015*(c*e*x+b*e-c*d)*(1155*c^4*e^4*g*x^4-840*b*c^3*e^4*g*x^3+4935*c^4*d*e^3*g*x^3+1365*c^4*e^4*f*x^3+560*b^
2*c^2*e^4*g*x^2-3850*b*c^3*d*e^3*g*x^2-910*b*c^3*e^4*f*x^2+8295*c^4*d^2*e^2*g*x^2+5915*c^4*d*e^3*f*x^2-320*b^3
*c*e^4*g*x+2520*b^2*c^2*d*e^3*g*x+520*b^2*c^2*e^4*f*x-6940*b*c^3*d^2*e^2*g*x-3900*b*c^3*d*e^3*f*x+6885*c^4*d^3
*e*g*x+9815*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1136*b^3*c*d*e^3*g-208*b^3*c*e^4*f+3784*b^2*c^2*d^2*e^2*g+1768*b^2*c
^2*d*e^3*f-5530*b*c^3*d^3*e*g-5486*b*c^3*d^2*e^2*f+2754*c^4*d^4*g+6929*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(3/2)/c^5/e^2/(e*x+d)^(3/2)

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maxima [B]  time = 0.99, size = 676, normalized size = 1.95 \begin {gather*} -\frac {2 \, {\left (105 \, c^{5} e^{5} x^{5} + 533 \, c^{5} d^{5} - 1488 \, b c^{4} d^{4} e + 1513 \, b^{2} c^{3} d^{3} e^{2} - 710 \, b^{3} c^{2} d^{2} e^{3} + 168 \, b^{4} c d e^{4} - 16 \, b^{5} e^{5} + 35 \, {\left (7 \, c^{5} d e^{4} + 4 \, b c^{4} e^{5}\right )} x^{4} - 5 \, {\left (10 \, c^{5} d^{2} e^{3} - 108 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} x^{3} - 3 \, {\left (174 \, c^{5} d^{3} e^{2} - 236 \, b c^{4} d^{2} e^{3} - 17 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (311 \, c^{5} d^{4} e - 100 \, b c^{4} d^{3} e^{2} - 279 \, b^{2} c^{3} d^{2} e^{3} + 76 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{1155 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} - \frac {2 \, {\left (1155 \, c^{6} e^{6} x^{6} + 2754 \, c^{6} d^{6} - 11038 \, b c^{5} d^{5} e + 17598 \, b^{2} c^{4} d^{4} e^{2} - 14234 \, b^{3} c^{3} d^{3} e^{3} + 6184 \, b^{4} c^{2} d^{2} e^{4} - 1392 \, b^{5} c d e^{5} + 128 \, b^{6} e^{6} + 105 \, {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} x^{5} - 35 \, {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} x^{4} - 5 \, {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} x^{3} - 3 \, {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} x^{2} + {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/1155*(105*c^5*e^5*x^5 + 533*c^5*d^5 - 1488*b*c^4*d^4*e + 1513*b^2*c^3*d^3*e^2 - 710*b^3*c^2*d^2*e^3 + 168*b
^4*c*d*e^4 - 16*b^5*e^5 + 35*(7*c^5*d*e^4 + 4*b*c^4*e^5)*x^4 - 5*(10*c^5*d^2*e^3 - 108*b*c^4*d*e^4 - b^2*c^3*e
^5)*x^3 - 3*(174*c^5*d^3*e^2 - 236*b*c^4*d^2*e^3 - 17*b^2*c^3*d*e^4 + 2*b^3*c^2*e^5)*x^2 - (311*c^5*d^4*e - 10
0*b*c^4*d^3*e^2 - 279*b^2*c^3*d^2*e^3 + 76*b^3*c^2*d*e^4 - 8*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*
f/(c^4*e^2*x + c^4*d*e) - 2/15015*(1155*c^6*e^6*x^6 + 2754*c^6*d^6 - 11038*b*c^5*d^5*e + 17598*b^2*c^4*d^4*e^2
- 14234*b^3*c^3*d^3*e^3 + 6184*b^4*c^2*d^2*e^4 - 1392*b^5*c*d*e^5 + 128*b^6*e^6 + 105*(25*c^6*d*e^5 + 14*b*c^
5*e^6)*x^5 - 35*(12*c^6*d^2*e^4 - 154*b*c^5*d*e^5 - b^2*c^4*e^6)*x^4 - 5*(954*c^6*d^3*e^3 - 1328*b*c^5*d^2*e^4
- 63*b^2*c^4*d*e^5 + 8*b^3*c^3*e^6)*x^3 - 3*(907*c^6*d^4*e^2 - 560*b*c^5*d^3*e^3 - 473*b^2*c^4*d^2*e^4 + 142*
b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*x^2 + (1377*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657*b^2*c^4*d^3*e^3 - 2460*b^3*c
^3*d^2*e^4 + 632*b^4*c^2*d*e^5 - 64*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^5*e^3*x + c^5*d*e^2)

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mupad [B]  time = 3.63, size = 637, normalized size = 1.84 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (14\,b\,e\,g+25\,c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,c\,e^3\,g\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^4\,c^2\,e^6-852\,g\,b^3\,c^3\,d\,e^5-156\,f\,b^3\,c^3\,e^6+2838\,g\,b^2\,c^4\,d^2\,e^4+1326\,f\,b^2\,c^4\,d\,e^5+3360\,g\,b\,c^5\,d^3\,e^3+18408\,f\,b\,c^5\,d^2\,e^4-5442\,g\,c^6\,d^4\,e^2-13572\,f\,c^6\,d^3\,e^3\right )}{15015\,c^5\,e^3}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1136\,g\,b^3\,c\,d\,e^3-208\,f\,b^3\,c\,e^4+3784\,g\,b^2\,c^2\,d^2\,e^2+1768\,f\,b^2\,c^2\,d\,e^3-5530\,g\,b\,c^3\,d^3\,e-5486\,f\,b\,c^3\,d^2\,e^2+2754\,g\,c^4\,d^4+6929\,f\,c^4\,d^3\,e\right )}{15015\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (g\,b^2\,e^2+154\,g\,b\,c\,d\,e+52\,f\,b\,c\,e^2-12\,g\,c^2\,d^2+91\,f\,c^2\,d\,e\right )}{429\,c}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,g\,b^3\,c^3\,e^6+630\,g\,b^2\,c^4\,d\,e^5+130\,f\,b^2\,c^4\,e^6+13280\,g\,b\,c^5\,d^2\,e^4+14040\,f\,b\,c^5\,d\,e^5-9540\,g\,c^6\,d^3\,e^3-1300\,f\,c^6\,d^2\,e^4\right )}{15015\,c^5\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-64\,g\,b^4\,e^4+568\,g\,b^3\,c\,d\,e^3+104\,f\,b^3\,c\,e^4-1892\,g\,b^2\,c^2\,d^2\,e^2-884\,f\,b^2\,c^2\,d\,e^3+2765\,g\,b\,c^3\,d^3\,e+2743\,f\,b\,c^3\,d^2\,e^2-1377\,g\,c^4\,d^4+4043\,f\,c^4\,d^3\,e\right )}{15015\,c^4\,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)^(3/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^2*x^5*(d + e*x)^(1/2)*(14*b*e*g + 25*c*d*g + 13*c*e*f))/14
3 + (2*c*e^3*g*x^6*(d + e*x)^(1/2))/13 + (x^2*(d + e*x)^(1/2)*(96*b^4*c^2*e^6*g - 156*b^3*c^3*e^6*f - 13572*c^
6*d^3*e^3*f - 5442*c^6*d^4*e^2*g + 18408*b*c^5*d^2*e^4*f + 1326*b^2*c^4*d*e^5*f + 3360*b*c^5*d^3*e^3*g - 852*b
^3*c^3*d*e^5*g + 2838*b^2*c^4*d^2*e^4*g))/(15015*c^5*e^3) + (2*(b*e - c*d)^2*(d + e*x)^(1/2)*(128*b^4*e^4*g +
2754*c^4*d^4*g - 208*b^3*c*e^4*f + 6929*c^4*d^3*e*f - 5530*b*c^3*d^3*e*g - 1136*b^3*c*d*e^3*g - 5486*b*c^3*d^2
*e^2*f + 1768*b^2*c^2*d*e^3*f + 3784*b^2*c^2*d^2*e^2*g))/(15015*c^5*e^3) + (2*e*x^4*(d + e*x)^(1/2)*(b^2*e^2*g
- 12*c^2*d^2*g + 52*b*c*e^2*f + 91*c^2*d*e*f + 154*b*c*d*e*g))/(429*c) + (x^3*(d + e*x)^(1/2)*(130*b^2*c^4*e^
6*f - 80*b^3*c^3*e^6*g - 1300*c^6*d^2*e^4*f - 9540*c^6*d^3*e^3*g + 14040*b*c^5*d*e^5*f + 13280*b*c^5*d^2*e^4*g
+ 630*b^2*c^4*d*e^5*g))/(15015*c^5*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(104*b^3*c*e^4*f - 1377*c^4*d^4*g
- 64*b^4*e^4*g + 4043*c^4*d^3*e*f + 2765*b*c^3*d^3*e*g + 568*b^3*c*d*e^3*g + 2743*b*c^3*d^2*e^2*f - 884*b^2*c^
2*d*e^3*f - 1892*b^2*c^2*d^2*e^2*g))/(15015*c^4*e^2)))/(x + d/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**(3/2)*(f + g*x), x)

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