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

Optimal. Leaf size=343 \[ -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \]

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Rubi [A]  time = 0.57, antiderivative size = 343, 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 {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(2*c*d - b*e)^3*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(45045*c^5*e^2*
(d + e*x)^(7/2)) - (16*(2*c*d - b*e)^2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2
))/(6435*c^4*e^2*(d + e*x)^(5/2)) - (4*(2*c*d - b*e)*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(7/2))/(715*c^3*e^2*(d + e*x)^(3/2)) - (2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(195*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(15*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 \sqrt {d+e x} (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 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\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}{15 c e^3}\\ &=-\frac {2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac {(2 (2 c d-b e) (15 c e f+c d g-8 b e g)) \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}{65 c^2 e}\\ &=-\frac {4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac {\left (8 (2 c d-b e)^2 (15 c e f+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 )^{5/2}}{(d+e x)^{3/2}} \, dx}{715 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac {4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac {\left (16 (2 c d-b e)^3 (15 c e f+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 )^{5/2}}{(d+e x)^{5/2}} \, dx}{6435 c^4 e}\\ &=-\frac {32 (2 c d-b e)^3 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac {16 (2 c d-b e)^2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac {4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 264, normalized size = 0.77 \begin {gather*} \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (77 d g+15 e f+28 e g x)+24 b^2 c^2 e^2 \left (187 d^2 g+d e (95 f+161 g x)+7 e^2 x (5 f+6 g x)\right )-2 b c^3 e \left (3611 d^3 g+d^2 e (4065 f+5922 g x)+21 d e^2 x (170 f+183 g x)+21 e^3 x^2 (45 f+44 g x)\right )+c^4 \left (3838 d^4 g+d^3 e (12525 f+13433 g x)+147 d^2 e^2 x (145 f+129 g x)+21 d e^3 x^2 (675 f+583 g x)+231 e^4 x^3 (15 f+13 g x)\right )\right )}{45045 c^5 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4*e^4*g - 16*b^3*c*e^3*(15*e*f + 77*
d*g + 28*e*g*x) + 24*b^2*c^2*e^2*(187*d^2*g + 7*e^2*x*(5*f + 6*g*x) + d*e*(95*f + 161*g*x)) - 2*b*c^3*e*(3611*
d^3*g + 21*e^3*x^2*(45*f + 44*g*x) + 21*d*e^2*x*(170*f + 183*g*x) + d^2*e*(4065*f + 5922*g*x)) + c^4*(3838*d^4
*g + 231*e^4*x^3*(15*f + 13*g*x) + 147*d^2*e^2*x*(145*f + 129*g*x) + 21*d*e^3*x^2*(675*f + 583*g*x) + d^3*e*(1
2525*f + 13433*g*x))))/(45045*c^5*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 7.43, size = 401, normalized size = 1.17 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{7/2} \left (128 b^4 e^4 g-448 b^3 c e^3 g (d+e x)-784 b^3 c d e^3 g-240 b^3 c e^4 f+1632 b^2 c^2 d^2 e^2 g+840 b^2 c^2 e^3 f (d+e x)+1440 b^2 c^2 d e^3 f+1008 b^2 c^2 e^2 g (d+e x)^2+1848 b^2 c^2 d e^2 g (d+e x)-1216 b c^3 d^3 e g-2880 b c^3 d^2 e^2 f-2016 b c^3 d^2 e g (d+e x)-1890 b c^3 e^2 f (d+e x)^2-3360 b c^3 d e^2 f (d+e x)-1848 b c^3 e g (d+e x)^3-2142 b c^3 d e g (d+e x)^2+128 c^4 d^4 g+1920 c^4 d^3 e f+224 c^4 d^3 g (d+e x)+3360 c^4 d^2 e f (d+e x)+252 c^4 d^2 g (d+e x)^2+3465 c^4 e f (d+e x)^3+3780 c^4 d e f (d+e x)^2+3003 c^4 g (d+e x)^4+231 c^4 d g (d+e x)^3\right )}{45045 c^5 e^2 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[d + e*x]*(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)*(1920*c^4*d^3*e*f - 2880*b*c^3*d^2*e^2*f + 1440*b^2*c^2*d*
e^3*f - 240*b^3*c*e^4*f + 128*c^4*d^4*g - 1216*b*c^3*d^3*e*g + 1632*b^2*c^2*d^2*e^2*g - 784*b^3*c*d*e^3*g + 12
8*b^4*e^4*g + 3360*c^4*d^2*e*f*(d + e*x) - 3360*b*c^3*d*e^2*f*(d + e*x) + 840*b^2*c^2*e^3*f*(d + e*x) + 224*c^
4*d^3*g*(d + e*x) - 2016*b*c^3*d^2*e*g*(d + e*x) + 1848*b^2*c^2*d*e^2*g*(d + e*x) - 448*b^3*c*e^3*g*(d + e*x)
+ 3780*c^4*d*e*f*(d + e*x)^2 - 1890*b*c^3*e^2*f*(d + e*x)^2 + 252*c^4*d^2*g*(d + e*x)^2 - 2142*b*c^3*d*e*g*(d
+ e*x)^2 + 1008*b^2*c^2*e^2*g*(d + e*x)^2 + 3465*c^4*e*f*(d + e*x)^3 + 231*c^4*d*g*(d + e*x)^3 - 1848*b*c^3*e*
g*(d + e*x)^3 + 3003*c^4*g*(d + e*x)^4))/(45045*c^5*e^2*(d + e*x)^(7/2))

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fricas [B]  time = 0.43, size = 881, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} e^{7} g x^{7} + 231 \, {\left (15 \, c^{7} e^{7} f + {\left (14 \, c^{7} d e^{6} + 31 \, b c^{6} e^{7}\right )} g\right )} x^{6} + 63 \, {\left (15 \, {\left (4 \, c^{7} d e^{6} + 9 \, b c^{6} e^{7}\right )} f - {\left (139 \, c^{7} d^{2} e^{5} - 263 \, b c^{6} d e^{6} - 71 \, b^{2} c^{5} e^{7}\right )} g\right )} x^{5} - 35 \, {\left (3 \, {\left (103 \, c^{7} d^{2} e^{5} - 193 \, b c^{6} d e^{6} - 53 \, b^{2} c^{5} e^{7}\right )} f + {\left (278 \, c^{7} d^{3} e^{4} + 54 \, b c^{6} d^{2} e^{5} - 474 \, b^{2} c^{5} d e^{6} - b^{3} c^{4} e^{7}\right )} g\right )} x^{4} - 5 \, {\left (3 \, {\left (824 \, c^{7} d^{3} e^{4} + 206 \, b c^{6} d^{2} e^{5} - 1454 \, b^{2} c^{5} d e^{6} - 5 \, b^{3} c^{4} e^{7}\right )} f - {\left (1637 \, c^{7} d^{4} e^{3} - 5930 \, b c^{6} d^{3} e^{4} + 4224 \, b^{2} c^{5} d^{2} e^{5} + 77 \, b^{3} c^{4} d e^{6} - 8 \, b^{4} c^{3} e^{7}\right )} g\right )} x^{3} + 3 \, {\left (15 \, {\left (271 \, c^{7} d^{4} e^{3} - 954 \, b c^{6} d^{3} e^{4} + 664 \, b^{2} c^{5} d^{2} e^{5} + 21 \, b^{3} c^{4} d e^{6} - 2 \, b^{4} c^{3} e^{7}\right )} f + {\left (3274 \, c^{7} d^{5} e^{2} - 6125 \, b c^{6} d^{4} e^{3} + 2290 \, b^{2} c^{5} d^{3} e^{4} + 715 \, b^{3} c^{4} d^{2} e^{5} - 170 \, b^{4} c^{3} d e^{6} + 16 \, b^{5} c^{2} e^{7}\right )} g\right )} x^{2} - 15 \, {\left (835 \, c^{7} d^{6} e - 3047 \, b c^{6} d^{5} e^{2} + 4283 \, b^{2} c^{5} d^{4} e^{3} - 2933 \, b^{3} c^{4} d^{3} e^{4} + 1046 \, b^{4} c^{3} d^{2} e^{5} - 200 \, b^{5} c^{2} d e^{6} + 16 \, b^{6} c e^{7}\right )} f - 2 \, {\left (1919 \, c^{7} d^{7} - 9368 \, b c^{6} d^{6} e + 18834 \, b^{2} c^{5} d^{5} e^{2} - 20100 \, b^{3} c^{4} d^{4} e^{3} + 12255 \, b^{4} c^{3} d^{3} e^{4} - 4284 \, b^{5} c^{2} d^{2} e^{5} + 808 \, b^{6} c d e^{6} - 64 \, b^{7} e^{7}\right )} g + {\left (15 \, {\left (1084 \, c^{7} d^{5} e^{2} - 1897 \, b c^{6} d^{4} e^{3} + 466 \, b^{2} c^{5} d^{3} e^{4} + 431 \, b^{3} c^{4} d^{2} e^{5} - 92 \, b^{4} c^{3} d e^{6} + 8 \, b^{5} c^{2} e^{7}\right )} f - {\left (1919 \, c^{7} d^{6} e - 7449 \, b c^{6} d^{5} e^{2} + 11385 \, b^{2} c^{5} d^{4} e^{3} - 8715 \, b^{3} c^{4} d^{3} e^{4} + 3540 \, b^{4} c^{3} d^{2} e^{5} - 744 \, b^{5} c^{2} d e^{6} + 64 \, b^{6} c e^{7}\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}}{45045 \, {\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)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + (14*c^7*d*e^6 + 31*b*c^6*e^7)*g)*x^6 + 63*(15*(4*c^7*d*e^6 +
9*b*c^6*e^7)*f - (139*c^7*d^2*e^5 - 263*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*g)*x^5 - 35*(3*(103*c^7*d^2*e^5 - 193*b
*c^6*d*e^6 - 53*b^2*c^5*e^7)*f + (278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^2*c^5*d*e^6 - b^3*c^4*e^7)*g)*x^4
- 5*(3*(824*c^7*d^3*e^4 + 206*b*c^6*d^2*e^5 - 1454*b^2*c^5*d*e^6 - 5*b^3*c^4*e^7)*f - (1637*c^7*d^4*e^3 - 593
0*b*c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*g)*x^3 + 3*(15*(271*c^7*d^4*e^3 - 9
54*b*c^6*d^3*e^4 + 664*b^2*c^5*d^2*e^5 + 21*b^3*c^4*d*e^6 - 2*b^4*c^3*e^7)*f + (3274*c^7*d^5*e^2 - 6125*b*c^6*
d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*g)*x^2 - 15*(835*c^
7*d^6*e - 3047*b*c^6*d^5*e^2 + 4283*b^2*c^5*d^4*e^3 - 2933*b^3*c^4*d^3*e^4 + 1046*b^4*c^3*d^2*e^5 - 200*b^5*c^
2*d*e^6 + 16*b^6*c*e^7)*f - 2*(1919*c^7*d^7 - 9368*b*c^6*d^6*e + 18834*b^2*c^5*d^5*e^2 - 20100*b^3*c^4*d^4*e^3
+ 12255*b^4*c^3*d^3*e^4 - 4284*b^5*c^2*d^2*e^5 + 808*b^6*c*d*e^6 - 64*b^7*e^7)*g + (15*(1084*c^7*d^5*e^2 - 18
97*b*c^6*d^4*e^3 + 466*b^2*c^5*d^3*e^4 + 431*b^3*c^4*d^2*e^5 - 92*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*f - (1919*c^7
*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*b^4*c^3*d^2*e^5 - 744*b^5*c^
2*d*e^6 + 64*b^6*c*e^7)*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 {5}{2}} \sqrt {e x + d} {\left (g x + f\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(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)*sqrt(e*x + d)*(g*x + f), x)

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maple [A]  time = 0.05, size = 367, normalized size = 1.07 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3003 g \,e^{4} x^{4} c^{4}-1848 b \,c^{3} e^{4} g \,x^{3}+12243 c^{4} d \,e^{3} g \,x^{3}+3465 c^{4} e^{4} f \,x^{3}+1008 b^{2} c^{2} e^{4} g \,x^{2}-7686 b \,c^{3} d \,e^{3} g \,x^{2}-1890 b \,c^{3} e^{4} f \,x^{2}+18963 c^{4} d^{2} e^{2} g \,x^{2}+14175 c^{4} d \,e^{3} f \,x^{2}-448 b^{3} c \,e^{4} g x +3864 b^{2} c^{2} d \,e^{3} g x +840 b^{2} c^{2} e^{4} f x -11844 b \,c^{3} d^{2} e^{2} g x -7140 b \,c^{3} d \,e^{3} f x +13433 c^{4} d^{3} e g x +21315 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1232 b^{3} c d \,e^{3} g -240 b^{3} c \,e^{4} f +4488 b^{2} c^{2} d^{2} e^{2} g +2280 b^{2} c^{2} d \,e^{3} f -7222 b \,c^{3} d^{3} e g -8130 b \,c^{3} d^{2} e^{2} f +3838 c^{4} d^{4} g +12525 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 {5}{2}}}{45045 \left (e x +d \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)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/45045*(c*e*x+b*e-c*d)*(3003*c^4*e^4*g*x^4-1848*b*c^3*e^4*g*x^3+12243*c^4*d*e^3*g*x^3+3465*c^4*e^4*f*x^3+1008
*b^2*c^2*e^4*g*x^2-7686*b*c^3*d*e^3*g*x^2-1890*b*c^3*e^4*f*x^2+18963*c^4*d^2*e^2*g*x^2+14175*c^4*d*e^3*f*x^2-4
48*b^3*c*e^4*g*x+3864*b^2*c^2*d*e^3*g*x+840*b^2*c^2*e^4*f*x-11844*b*c^3*d^2*e^2*g*x-7140*b*c^3*d*e^3*f*x+13433
*c^4*d^3*e*g*x+21315*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1232*b^3*c*d*e^3*g-240*b^3*c*e^4*f+4488*b^2*c^2*d^2*e^2*g+2
280*b^2*c^2*d*e^3*f-7222*b*c^3*d^3*e*g-8130*b*c^3*d^2*e^2*f+3838*c^4*d^4*g+12525*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(5/2)/c^5/e^2/(e*x+d)^(5/2)

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maxima [B]  time = 1.00, size = 878, normalized size = 2.56 \begin {gather*} \frac {2 \, {\left (231 \, c^{6} e^{6} x^{6} - 835 \, c^{6} d^{6} + 3047 \, b c^{5} d^{5} e - 4283 \, b^{2} c^{4} d^{4} e^{2} + 2933 \, b^{3} c^{3} d^{3} e^{3} - 1046 \, b^{4} c^{2} d^{2} e^{4} + 200 \, b^{5} c d e^{5} - 16 \, b^{6} e^{6} + 63 \, {\left (4 \, c^{6} d e^{5} + 9 \, b c^{5} e^{6}\right )} x^{5} - 7 \, {\left (103 \, c^{6} d^{2} e^{4} - 193 \, b c^{5} d e^{5} - 53 \, b^{2} c^{4} e^{6}\right )} x^{4} - {\left (824 \, c^{6} d^{3} e^{3} + 206 \, b c^{5} d^{2} e^{4} - 1454 \, b^{2} c^{4} d e^{5} - 5 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \, {\left (271 \, c^{6} d^{4} e^{2} - 954 \, b c^{5} d^{3} e^{3} + 664 \, b^{2} c^{4} d^{2} e^{4} + 21 \, b^{3} c^{3} d e^{5} - 2 \, b^{4} c^{2} e^{6}\right )} x^{2} + {\left (1084 \, c^{6} d^{5} e - 1897 \, b c^{5} d^{4} e^{2} + 466 \, b^{2} c^{4} d^{3} e^{3} + 431 \, b^{3} c^{3} d^{2} e^{4} - 92 \, b^{4} c^{2} d e^{5} + 8 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{3003 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac {2 \, {\left (3003 \, c^{7} e^{7} x^{7} - 3838 \, c^{7} d^{7} + 18736 \, b c^{6} d^{6} e - 37668 \, b^{2} c^{5} d^{5} e^{2} + 40200 \, b^{3} c^{4} d^{4} e^{3} - 24510 \, b^{4} c^{3} d^{3} e^{4} + 8568 \, b^{5} c^{2} d^{2} e^{5} - 1616 \, b^{6} c d e^{6} + 128 \, b^{7} e^{7} + 231 \, {\left (14 \, c^{7} d e^{6} + 31 \, b c^{6} e^{7}\right )} x^{6} - 63 \, {\left (139 \, c^{7} d^{2} e^{5} - 263 \, b c^{6} d e^{6} - 71 \, b^{2} c^{5} e^{7}\right )} x^{5} - 35 \, {\left (278 \, c^{7} d^{3} e^{4} + 54 \, b c^{6} d^{2} e^{5} - 474 \, b^{2} c^{5} d e^{6} - b^{3} c^{4} e^{7}\right )} x^{4} + 5 \, {\left (1637 \, c^{7} d^{4} e^{3} - 5930 \, b c^{6} d^{3} e^{4} + 4224 \, b^{2} c^{5} d^{2} e^{5} + 77 \, b^{3} c^{4} d e^{6} - 8 \, b^{4} c^{3} e^{7}\right )} x^{3} + 3 \, {\left (3274 \, c^{7} d^{5} e^{2} - 6125 \, b c^{6} d^{4} e^{3} + 2290 \, b^{2} c^{5} d^{3} e^{4} + 715 \, b^{3} c^{4} d^{2} e^{5} - 170 \, b^{4} c^{3} d e^{6} + 16 \, b^{5} c^{2} e^{7}\right )} x^{2} - {\left (1919 \, c^{7} d^{6} e - 7449 \, b c^{6} d^{5} e^{2} + 11385 \, b^{2} c^{5} d^{4} e^{3} - 8715 \, b^{3} c^{4} d^{3} e^{4} + 3540 \, b^{4} c^{3} d^{2} e^{5} - 744 \, b^{5} c^{2} d e^{6} + 64 \, b^{6} c e^{7}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{45045 \, {\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)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*c^6*e^6*x^6 - 835*c^6*d^6 + 3047*b*c^5*d^5*e - 4283*b^2*c^4*d^4*e^2 + 2933*b^3*c^3*d^3*e^3 - 1046*
b^4*c^2*d^2*e^4 + 200*b^5*c*d*e^5 - 16*b^6*e^6 + 63*(4*c^6*d*e^5 + 9*b*c^5*e^6)*x^5 - 7*(103*c^6*d^2*e^4 - 193
*b*c^5*d*e^5 - 53*b^2*c^4*e^6)*x^4 - (824*c^6*d^3*e^3 + 206*b*c^5*d^2*e^4 - 1454*b^2*c^4*d*e^5 - 5*b^3*c^3*e^6
)*x^3 + 3*(271*c^6*d^4*e^2 - 954*b*c^5*d^3*e^3 + 664*b^2*c^4*d^2*e^4 + 21*b^3*c^3*d*e^5 - 2*b^4*c^2*e^6)*x^2 +
(1084*c^6*d^5*e - 1897*b*c^5*d^4*e^2 + 466*b^2*c^4*d^3*e^3 + 431*b^3*c^3*d^2*e^4 - 92*b^4*c^2*d*e^5 + 8*b^5*c
*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^4*e^2*x + c^4*d*e) + 2/45045*(3003*c^7*e^7*x^7 - 3838*c^7*d^7
+ 18736*b*c^6*d^6*e - 37668*b^2*c^5*d^5*e^2 + 40200*b^3*c^4*d^4*e^3 - 24510*b^4*c^3*d^3*e^4 + 8568*b^5*c^2*d^
2*e^5 - 1616*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(14*c^7*d*e^6 + 31*b*c^6*e^7)*x^6 - 63*(139*c^7*d^2*e^5 - 263*b*c
^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^2*c^5*d*e^6 - b^3*c^4*e^7)*x^4
+ 5*(1637*c^7*d^4*e^3 - 5930*b*c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*x^3 + 3
*(3274*c^7*d^5*e^2 - 6125*b*c^6*d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*
b^5*c^2*e^7)*x^2 - (1919*c^7*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*
b^4*c^3*d^2*e^5 - 744*b^5*c^2*d*e^6 + 64*b^6*c*e^7)*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 = 4.03, size = 769, normalized size = 2.24 \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 (71\,g\,b^2\,e^2+263\,g\,b\,c\,d\,e+135\,f\,b\,c\,e^2-139\,g\,c^2\,d^2+60\,f\,c^2\,d\,e\right )}{715}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,g\,b^4\,c^3\,e^7+770\,g\,b^3\,c^4\,d\,e^6+150\,f\,b^3\,c^4\,e^7+42240\,g\,b^2\,c^5\,d^2\,e^5+43620\,f\,b^2\,c^5\,d\,e^6-59300\,g\,b\,c^6\,d^3\,e^4-6180\,f\,b\,c^6\,d^2\,e^5+16370\,g\,c^7\,d^4\,e^3-24720\,f\,c^7\,d^3\,e^4\right )}{45045\,c^5\,e^3}+\frac {2\,c^2\,e^4\,g\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1232\,g\,b^3\,c\,d\,e^3-240\,f\,b^3\,c\,e^4+4488\,g\,b^2\,c^2\,d^2\,e^2+2280\,f\,b^2\,c^2\,d\,e^3-7222\,g\,b\,c^3\,d^3\,e-8130\,f\,b\,c^3\,d^2\,e^2+3838\,g\,c^4\,d^4+12525\,f\,c^4\,d^3\,e\right )}{45045\,c^5\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (70\,g\,b^3\,c^4\,e^7+33180\,g\,b^2\,c^5\,d\,e^6+11130\,f\,b^2\,c^5\,e^7-3780\,g\,b\,c^6\,d^2\,e^5+40530\,f\,b\,c^6\,d\,e^6-19460\,g\,c^7\,d^3\,e^4-21630\,f\,c^7\,d^2\,e^5\right )}{45045\,c^5\,e^3}+\frac {2\,c\,e^3\,x^6\,\sqrt {d+e\,x}\,\left (31\,b\,e\,g+14\,c\,d\,g+15\,c\,e\,f\right )}{195}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (16\,g\,b^4\,e^4-154\,g\,b^3\,c\,d\,e^3-30\,f\,b^3\,c\,e^4+561\,g\,b^2\,c^2\,d^2\,e^2+285\,f\,b^2\,c^2\,d\,e^3+2851\,g\,b\,c^3\,d^3\,e+10245\,f\,b\,c^3\,d^2\,e^2-3274\,g\,c^4\,d^4-4065\,f\,c^4\,d^3\,e\right )}{15015\,c^3\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-64\,g\,b^4\,e^4+616\,g\,b^3\,c\,d\,e^3+120\,f\,b^3\,c\,e^4-2244\,g\,b^2\,c^2\,d^2\,e^2-1140\,f\,b^2\,c^2\,d\,e^3+3611\,g\,b\,c^3\,d^3\,e+4065\,f\,b\,c^3\,d^2\,e^2-1919\,g\,c^4\,d^4+16260\,f\,c^4\,d^3\,e\right )}{45045\,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)^(1/2)*(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^2*x^5*(d + e*x)^(1/2)*(71*b^2*e^2*g - 139*c^2*d^2*g + 135*b
*c*e^2*f + 60*c^2*d*e*f + 263*b*c*d*e*g))/715 + (x^3*(d + e*x)^(1/2)*(150*b^3*c^4*e^7*f - 80*b^4*c^3*e^7*g - 2
4720*c^7*d^3*e^4*f + 16370*c^7*d^4*e^3*g - 6180*b*c^6*d^2*e^5*f + 43620*b^2*c^5*d*e^6*f - 59300*b*c^6*d^3*e^4*
g + 770*b^3*c^4*d*e^6*g + 42240*b^2*c^5*d^2*e^5*g))/(45045*c^5*e^3) + (2*c^2*e^4*g*x^7*(d + e*x)^(1/2))/15 + (
2*(b*e - c*d)^3*(d + e*x)^(1/2)*(128*b^4*e^4*g + 3838*c^4*d^4*g - 240*b^3*c*e^4*f + 12525*c^4*d^3*e*f - 7222*b
*c^3*d^3*e*g - 1232*b^3*c*d*e^3*g - 8130*b*c^3*d^2*e^2*f + 2280*b^2*c^2*d*e^3*f + 4488*b^2*c^2*d^2*e^2*g))/(45
045*c^5*e^3) + (x^4*(d + e*x)^(1/2)*(11130*b^2*c^5*e^7*f + 70*b^3*c^4*e^7*g - 21630*c^7*d^2*e^5*f - 19460*c^7*
d^3*e^4*g + 40530*b*c^6*d*e^6*f - 3780*b*c^6*d^2*e^5*g + 33180*b^2*c^5*d*e^6*g))/(45045*c^5*e^3) + (2*c*e^3*x^
6*(d + e*x)^(1/2)*(31*b*e*g + 14*c*d*g + 15*c*e*f))/195 + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(16*b^4*e^4*g - 3
274*c^4*d^4*g - 30*b^3*c*e^4*f - 4065*c^4*d^3*e*f + 2851*b*c^3*d^3*e*g - 154*b^3*c*d*e^3*g + 10245*b*c^3*d^2*e
^2*f + 285*b^2*c^2*d*e^3*f + 561*b^2*c^2*d^2*e^2*g))/(15015*c^3*e) + (2*x*(b*e - c*d)^2*(d + e*x)^(1/2)*(120*b
^3*c*e^4*f - 1919*c^4*d^4*g - 64*b^4*e^4*g + 16260*c^4*d^3*e*f + 3611*b*c^3*d^3*e*g + 616*b^3*c*d*e^3*g + 4065
*b*c^3*d^2*e^2*f - 1140*b^2*c^2*d*e^3*f - 2244*b^2*c^2*d^2*e^2*g))/(45045*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 {5}{2}} \sqrt {d + e x} \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

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

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