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

Optimal. Leaf size=347 \[ -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 c^3 e^2}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2} \]

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Rubi [A]  time = 0.62, 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 )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 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 )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 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 )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 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 )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 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 )^{3/2}}{11 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(2*c*d - b*e)^3*(11*c*e*f + 5*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3465*c^5*e^2
*(d + e*x)^(3/2)) - (16*(2*c*d - b*e)^2*(11*c*e*f + 5*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(
3/2))/(1155*c^4*e^2*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(11*c*e*f + 5*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)^(3/2))/(231*c^3*e^2) - (2*(11*c*e*f + 5*c*d*g - 8*b*e*g)*(d + e*x)^(3/2)*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(99*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)^(
3/2))/(11*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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^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 )^{3/2}}{11 c e^2}-\frac {\left (2 \left (\frac {3}{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} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{11 c e^3}\\ &=-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 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 )^{3/2}}{11 c e^2}+\frac {(2 (2 c d-b e) (11 c e f+5 c d g-8 b e g)) \int (d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{33 c^2 e}\\ &=-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 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 )^{3/2}}{11 c e^2}+\frac {\left (8 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g)\right ) \int \sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{231 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 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 )^{3/2}}{11 c e^2}+\frac {\left (16 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{1155 c^4 e}\\ &=-\frac {32 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 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 )^{3/2}}{11 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 262, normalized size = 0.76 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (65 d g+11 e f+12 e g x)+24 b^2 c^2 e^2 \left (131 d^2 g+d e (55 f+57 g x)+e^2 x (11 f+10 g x)\right )-2 b c^3 e \left (2071 d^3 g+3 d^2 e (583 f+558 g x)+3 d e^2 x (286 f+245 g x)+5 e^3 x^2 (33 f+28 g x)\right )+c^4 \left (1910 d^4 g+d^3 e (3509 f+2865 g x)+3 d^2 e^2 x (1177 f+905 g x)+5 d e^3 x^2 (363 f+287 g x)+35 e^4 x^3 (11 f+9 g x)\right )\right )}{3465 c^5 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4*e^4*g - 16*b^3*c*e^3*(11*e*f + 65*d*
g + 12*e*g*x) + 24*b^2*c^2*e^2*(131*d^2*g + e^2*x*(11*f + 10*g*x) + d*e*(55*f + 57*g*x)) - 2*b*c^3*e*(2071*d^3
*g + 5*e^3*x^2*(33*f + 28*g*x) + 3*d*e^2*x*(286*f + 245*g*x) + 3*d^2*e*(583*f + 558*g*x)) + c^4*(1910*d^4*g +
35*e^4*x^3*(11*f + 9*g*x) + 5*d*e^3*x^2*(363*f + 287*g*x) + 3*d^2*e^2*x*(1177*f + 905*g*x) + d^3*e*(3509*f + 2
865*g*x))))/(3465*c^5*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.88, 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 )^{3/2} \left (128 b^4 e^4 g-192 b^3 c e^3 g (d+e x)-848 b^3 c d e^3 g-176 b^3 c e^4 f+2016 b^2 c^2 d^2 e^2 g+264 b^2 c^2 e^3 f (d+e x)+1056 b^2 c^2 d e^3 f+240 b^2 c^2 e^2 g (d+e x)^2+888 b^2 c^2 d e^2 g (d+e x)-1984 b c^3 d^3 e g-2112 b c^3 d^2 e^2 f-1248 b c^3 d^2 e g (d+e x)-330 b c^3 e^2 f (d+e x)^2-1056 b c^3 d e^2 f (d+e x)-280 b c^3 e g (d+e x)^3-630 b c^3 d e g (d+e x)^2+640 c^4 d^4 g+1408 c^4 d^3 e f+480 c^4 d^3 g (d+e x)+1056 c^4 d^2 e f (d+e x)+300 c^4 d^2 g (d+e x)^2+385 c^4 e f (d+e x)^3+660 c^4 d e f (d+e x)^2+315 c^4 g (d+e x)^4+175 c^4 d g (d+e x)^3\right )}{3465 c^5 e^2 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(3/2)*(1408*c^4*d^3*e*f - 2112*b*c^3*d^2*e^2*f + 1056*b^2*c^2*d*
e^3*f - 176*b^3*c*e^4*f + 640*c^4*d^4*g - 1984*b*c^3*d^3*e*g + 2016*b^2*c^2*d^2*e^2*g - 848*b^3*c*d*e^3*g + 12
8*b^4*e^4*g + 1056*c^4*d^2*e*f*(d + e*x) - 1056*b*c^3*d*e^2*f*(d + e*x) + 264*b^2*c^2*e^3*f*(d + e*x) + 480*c^
4*d^3*g*(d + e*x) - 1248*b*c^3*d^2*e*g*(d + e*x) + 888*b^2*c^2*d*e^2*g*(d + e*x) - 192*b^3*c*e^3*g*(d + e*x) +
660*c^4*d*e*f*(d + e*x)^2 - 330*b*c^3*e^2*f*(d + e*x)^2 + 300*c^4*d^2*g*(d + e*x)^2 - 630*b*c^3*d*e*g*(d + e*
x)^2 + 240*b^2*c^2*e^2*g*(d + e*x)^2 + 385*c^4*e*f*(d + e*x)^3 + 175*c^4*d*g*(d + e*x)^3 - 280*b*c^3*e*g*(d +
e*x)^3 + 315*c^4*g*(d + e*x)^4))/(3465*c^5*e^2*(d + e*x)^(3/2))

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fricas [A]  time = 0.49, size = 499, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (11 \, {\left (26 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} f + {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 3 \, {\left (11 \, {\left (52 \, c^{5} d^{2} e^{3} + 13 \, b c^{4} d e^{4} - 2 \, b^{2} c^{3} e^{5}\right )} f + {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (319 \, c^{5} d^{4} e - 637 \, b c^{4} d^{3} e^{2} + 438 \, b^{2} c^{3} d^{2} e^{3} - 136 \, b^{3} c^{2} d e^{4} + 16 \, b^{4} c e^{5}\right )} f - 2 \, {\left (955 \, c^{5} d^{5} - 3026 \, b c^{4} d^{4} e + 3643 \, b^{2} c^{3} d^{3} e^{2} - 2092 \, b^{3} c^{2} d^{2} e^{3} + 584 \, b^{4} c d e^{4} - 64 \, b^{5} e^{5}\right )} g - {\left (11 \, {\left (2 \, c^{5} d^{3} e^{2} - 159 \, b c^{4} d^{2} e^{3} + 60 \, b^{2} c^{3} d e^{4} - 8 \, b^{3} c^{2} e^{5}\right )} f + {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3465 \, {\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)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*c^5*e^5*g*x^5 + 35*(11*c^5*e^5*f + (32*c^5*d*e^4 + b*c^4*e^5)*g)*x^4 + 5*(11*(26*c^5*d*e^4 + b*c^4
*e^5)*f + (256*c^5*d^2*e^3 + 49*b*c^4*d*e^4 - 8*b^2*c^3*e^5)*g)*x^3 + 3*(11*(52*c^5*d^2*e^3 + 13*b*c^4*d*e^4 -
2*b^2*c^3*e^5)*f + (50*c^5*d^3*e^2 + 279*b*c^4*d^2*e^3 - 114*b^2*c^3*d*e^4 + 16*b^3*c^2*e^5)*g)*x^2 - 11*(319
*c^5*d^4*e - 637*b*c^4*d^3*e^2 + 438*b^2*c^3*d^2*e^3 - 136*b^3*c^2*d*e^4 + 16*b^4*c*e^5)*f - 2*(955*c^5*d^5 -
3026*b*c^4*d^4*e + 3643*b^2*c^3*d^3*e^2 - 2092*b^3*c^2*d^2*e^3 + 584*b^4*c*d*e^4 - 64*b^5*e^5)*g - (11*(2*c^5*
d^3*e^2 - 159*b*c^4*d^2*e^3 + 60*b^2*c^3*d*e^4 - 8*b^3*c^2*e^5)*f + (955*c^5*d^4*e - 2071*b*c^4*d^3*e^2 + 1572
*b^2*c^3*d^2*e^3 - 520*b^3*c^2*d*e^4 + 64*b^4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x
+ d)/(c^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 \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\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)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^(5/2)*(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 (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 f \,d^{3} c^{4} e \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{3465 \sqrt {e x +d}\, c^{5} 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)^(1/2),x)

[Out]

2/3465*(c*e*x+b*e-c*d)*(315*c^4*e^4*g*x^4-280*b*c^3*e^4*g*x^3+1435*c^4*d*e^3*g*x^3+385*c^4*e^4*f*x^3+240*b^2*c
^2*e^4*g*x^2-1470*b*c^3*d*e^3*g*x^2-330*b*c^3*e^4*f*x^2+2715*c^4*d^2*e^2*g*x^2+1815*c^4*d*e^3*f*x^2-192*b^3*c*
e^4*g*x+1368*b^2*c^2*d*e^3*g*x+264*b^2*c^2*e^4*f*x-3348*b*c^3*d^2*e^2*g*x-1716*b*c^3*d*e^3*f*x+2865*c^4*d^3*e*
g*x+3531*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1040*b^3*c*d*e^3*g-176*b^3*c*e^4*f+3144*b^2*c^2*d^2*e^2*g+1320*b^2*c^2*
d*e^3*f-4142*b*c^3*d^3*e*g-3498*b*c^3*d^2*e^2*f+1910*c^4*d^4*g+3509*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)/c^5/e^2/(e*x+d)^(1/2)

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maxima [A]  time = 1.02, size = 501, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} - 319 \, c^{4} d^{4} + 637 \, b c^{3} d^{3} e - 438 \, b^{2} c^{2} d^{2} e^{2} + 136 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \, {\left (26 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (52 \, c^{4} d^{2} e^{2} + 13 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} - {\left (2 \, c^{4} d^{3} e - 159 \, b c^{3} d^{2} e^{2} + 60 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} - 1910 \, c^{5} d^{5} + 6052 \, b c^{4} d^{4} e - 7286 \, b^{2} c^{3} d^{3} e^{2} + 4184 \, b^{3} c^{2} d^{2} e^{3} - 1168 \, b^{4} c d e^{4} + 128 \, b^{5} e^{5} + 35 \, {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} x^{4} + 5 \, {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\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)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*c^4*e^4*x^4 - 319*c^4*d^4 + 637*b*c^3*d^3*e - 438*b^2*c^2*d^2*e^2 + 136*b^3*c*d*e^3 - 16*b^4*e^4 + 5
*(26*c^4*d*e^3 + b*c^3*e^4)*x^3 + 3*(52*c^4*d^2*e^2 + 13*b*c^3*d*e^3 - 2*b^2*c^2*e^4)*x^2 - (2*c^4*d^3*e - 159
*b*c^3*d^2*e^2 + 60*b^2*c^2*d*e^3 - 8*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^4*e^2*x + c^4*d*e)
+ 2/3465*(315*c^5*e^5*x^5 - 1910*c^5*d^5 + 6052*b*c^4*d^4*e - 7286*b^2*c^3*d^3*e^2 + 4184*b^3*c^2*d^2*e^3 - 1
168*b^4*c*d*e^4 + 128*b^5*e^5 + 35*(32*c^5*d*e^4 + b*c^4*e^5)*x^4 + 5*(256*c^5*d^2*e^3 + 49*b*c^4*d*e^4 - 8*b^
2*c^3*e^5)*x^3 + 3*(50*c^5*d^3*e^2 + 279*b*c^4*d^2*e^3 - 114*b^2*c^3*d*e^4 + 16*b^3*c^2*e^5)*x^2 - (955*c^5*d^
4*e - 2071*b*c^4*d^3*e^2 + 1572*b^2*c^3*d^2*e^3 - 520*b^3*c^2*d*e^4 + 64*b^4*c*e^5)*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 = 3.09, size = 501, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,e^2+49\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2+256\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2}+\frac {2\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1040\,g\,b^3\,c\,d\,e^3-176\,f\,b^3\,c\,e^4+3144\,g\,b^2\,c^2\,d^2\,e^2+1320\,f\,b^2\,c^2\,d\,e^3-4142\,g\,b\,c^3\,d^3\,e-3498\,f\,b\,c^3\,d^2\,e^2+1910\,g\,c^4\,d^4+3509\,f\,c^4\,d^3\,e\right )}{3465\,c^5\,e^3}-\frac {x\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,c\,e^5-1040\,g\,b^3\,c^2\,d\,e^4-176\,f\,b^3\,c^2\,e^5+3144\,g\,b^2\,c^3\,d^2\,e^3+1320\,f\,b^2\,c^3\,d\,e^4-4142\,g\,b\,c^4\,d^3\,e^2-3498\,f\,b\,c^4\,d^2\,e^3+1910\,g\,c^5\,d^4\,e+44\,f\,c^5\,d^3\,e^2\right )}{3465\,c^5\,e^3}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^3\,c^2\,e^5-684\,g\,b^2\,c^3\,d\,e^4-132\,f\,b^2\,c^3\,e^5+1674\,g\,b\,c^4\,d^2\,e^3+858\,f\,b\,c^4\,d\,e^4+300\,g\,c^5\,d^3\,e^2+3432\,f\,c^5\,d^2\,e^3\right )}{3465\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (b\,e\,g+32\,c\,d\,g+11\,c\,e\,f\right )}{99\,c}\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)^(1/2),x)

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*x^3*(d + e*x)^(1/2)*(256*c^2*d^2*g - 8*b^2*e^2*g + 11*b*c*e^2
*f + 286*c^2*d*e*f + 49*b*c*d*e*g))/(693*c^2) + (2*e^2*g*x^5*(d + e*x)^(1/2))/11 + (2*(b*e - c*d)*(d + e*x)^(1
/2)*(128*b^4*e^4*g + 1910*c^4*d^4*g - 176*b^3*c*e^4*f + 3509*c^4*d^3*e*f - 4142*b*c^3*d^3*e*g - 1040*b^3*c*d*e
^3*g - 3498*b*c^3*d^2*e^2*f + 1320*b^2*c^2*d*e^3*f + 3144*b^2*c^2*d^2*e^2*g))/(3465*c^5*e^3) - (x*(d + e*x)^(1
/2)*(44*c^5*d^3*e^2*f - 176*b^3*c^2*e^5*f + 128*b^4*c*e^5*g + 1910*c^5*d^4*e*g - 3498*b*c^4*d^2*e^3*f + 1320*b
^2*c^3*d*e^4*f - 4142*b*c^4*d^3*e^2*g - 1040*b^3*c^2*d*e^4*g + 3144*b^2*c^3*d^2*e^3*g))/(3465*c^5*e^3) + (x^2*
(d + e*x)^(1/2)*(96*b^3*c^2*e^5*g - 132*b^2*c^3*e^5*f + 3432*c^5*d^2*e^3*f + 300*c^5*d^3*e^2*g + 858*b*c^4*d*e
^4*f + 1674*b*c^4*d^2*e^3*g - 684*b^2*c^3*d*e^4*g))/(3465*c^5*e^3) + (2*e*x^4*(d + e*x)^(1/2)*(b*e*g + 32*c*d*
g + 11*c*e*f))/(99*c)))/(x + d/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \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)**(1/2),x)

[Out]

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

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