3.366 \(\int \frac {f+g x+h x^2+i x^3+j x^4}{(a+b x-c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=353 \[ \frac {2 \left (x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )+b^2 c (4 a j+b i)+c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j-a c h+c^2 f\right )+a b^3 j+a b^2 c i+2 a c^2 (a i+c g)\right )}{3 c^3 \left (4 a c+b^2\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )-b^2 c (28 a j+b i)+8 c^3 (b g-a h)-4 b^4 j+16 c^4 f\right )+4 b c^2 \left (8 a^2 j-a c h+2 c^2 f\right )+24 a^2 c^3 i+b^3 c (10 a j+c h)+2 b^2 c^2 (3 a i+2 c g)+b^5 j+b^4 c i\right )}{3 c^3 \left (4 a c+b^2\right )^2 \sqrt {a+b x-c x^2}}-\frac {j \tan ^{-1}\left (\frac {b-2 c x}{2 \sqrt {c} \sqrt {a+b x-c x^2}}\right )}{c^{5/2}} \]

[Out]

2/3*(a*b^2*c*i+2*a*c^2*(a*i+c*g)+a*b^3*j-b*c*(-3*a^2*j-a*c*h+c^2*f)+(2*c^4*f+c^3*(2*a*h+b*g)+b^4*j+b^2*c*(4*a*
j+b*i)+c^2*(2*a^2*j+3*a*b*i+b^2*h))*x)/c^3/(4*a*c+b^2)/(-c*x^2+b*x+a)^(3/2)-j*arctan(1/2*(-2*c*x+b)/c^(1/2)/(-
c*x^2+b*x+a)^(1/2))/c^(5/2)-2/3*(b^4*c*i+24*a^2*c^3*i+2*b^2*c^2*(3*a*i+2*c*g)+b^5*j+b^3*c*(10*a*j+c*h)+4*b*c^2
*(8*a^2*j-a*c*h+2*c^2*f)-c*(16*c^4*f+8*c^3*(-a*h+b*g)-4*b^4*j-b^2*c*(28*a*j+b*i)+2*c^2*(-16*a^2*j-6*a*b*i+b^2*
h))*x)/c^3/(4*a*c+b^2)^2/(-c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.39, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1660, 12, 621, 204} \[ -\frac {2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )-b^2 c (28 a j+b i)+8 c^3 (b g-a h)-4 b^4 j+16 c^4 f\right )+4 b c^2 \left (8 a^2 j-a c h+2 c^2 f\right )+24 a^2 c^3 i+2 b^2 c^2 (3 a i+2 c g)+b^3 c (10 a j+c h)+b^4 c i+b^5 j\right )}{3 c^3 \left (4 a c+b^2\right )^2 \sqrt {a+b x-c x^2}}+\frac {2 \left (x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )+b^2 c (4 a j+b i)+c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j-a c h+c^2 f\right )+a b^2 c i+a b^3 j+2 a c^2 (a i+c g)\right )}{3 c^3 \left (4 a c+b^2\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {j \tan ^{-1}\left (\frac {b-2 c x}{2 \sqrt {c} \sqrt {a+b x-c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x - c*x^2)^(5/2),x]

[Out]

(2*(a*b^2*c*i + 2*a*c^2*(c*g + a*i) + a*b^3*j - b*c*(c^2*f - a*c*h - 3*a^2*j) + (2*c^4*f + c^3*(b*g + 2*a*h) +
b^4*j + b^2*c*(b*i + 4*a*j) + c^2*(b^2*h + 3*a*b*i + 2*a^2*j))*x))/(3*c^3*(b^2 + 4*a*c)*(a + b*x - c*x^2)^(3/
2)) - (2*(b^4*c*i + 24*a^2*c^3*i + 2*b^2*c^2*(2*c*g + 3*a*i) + b^5*j + b^3*c*(c*h + 10*a*j) + 4*b*c^2*(2*c^2*f
- a*c*h + 8*a^2*j) - c*(16*c^4*f + 8*c^3*(b*g - a*h) - 4*b^4*j - b^2*c*(b*i + 28*a*j) + 2*c^2*(b^2*h - 6*a*b*
i - 16*a^2*j))*x))/(3*c^3*(b^2 + 4*a*c)^2*Sqrt[a + b*x - c*x^2]) - (j*ArcTan[(b - 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x - c*x^2])])/c^(5/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin {align*} \int \frac {f+g x+h x^2+366 x^3+j x^4}{\left (a+b x-c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (c^3 \left (b f-\frac {3 a^2 (244 c+b j)}{c^2}-\frac {a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {366 b^3 c+4 b c^3 g+b^4 j+b^2 c (c h+a j)+4 c^2 \left (2 c^2 f-a c h-a^2 j\right )}{2 c^3}+\frac {3 \left (b^2+4 a c\right ) (366 c+b j) x}{2 c^2}+\frac {3 \left (b^2+4 a c\right ) j x^2}{2 c}}{\left (a+b x-c x^2\right )^{3/2}} \, dx}{3 \left (b^2+4 a c\right )}\\ &=-\frac {2 \left (c^3 \left (b f-\frac {3 a^2 (244 c+b j)}{c^2}-\frac {a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt {a+b x-c x^2}}+\frac {4 \int \frac {3 \left (b^2+4 a c\right )^2 j}{4 c^2 \sqrt {a+b x-c x^2}} \, dx}{3 \left (b^2+4 a c\right )^2}\\ &=-\frac {2 \left (c^3 \left (b f-\frac {3 a^2 (244 c+b j)}{c^2}-\frac {a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt {a+b x-c x^2}}+\frac {j \int \frac {1}{\sqrt {a+b x-c x^2}} \, dx}{c^2}\\ &=-\frac {2 \left (c^3 \left (b f-\frac {3 a^2 (244 c+b j)}{c^2}-\frac {a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt {a+b x-c x^2}}+\frac {(2 j) \operatorname {Subst}\left (\int \frac {1}{-4 c-x^2} \, dx,x,\frac {b-2 c x}{\sqrt {a+b x-c x^2}}\right )}{c^2}\\ &=-\frac {2 \left (c^3 \left (b f-\frac {3 a^2 (244 c+b j)}{c^2}-\frac {a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac {2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt {a+b x-c x^2}}-\frac {j \tan ^{-1}\left (\frac {b-2 c x}{2 \sqrt {c} \sqrt {a+b x-c x^2}}\right )}{c^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 1.11, size = 319, normalized size = 0.90 \[ -\frac {2 \left (b^3 \left (3 a^2 j+18 a c j x^2+c^2 \left (f+3 g x-\left (x^2 (3 h+i x)\right )\right )\right )+2 b^2 c \left (21 a^2 j x+a c \left (g+x \left (-6 h+3 i x-14 j x^2\right )\right )+c^2 x (3 f+x (h x-6 g))\right )+4 b c \left (5 a^3 j-2 a^2 c (h-3 i x)+3 a c^2 \left (f-x \left (g-h x+i x^2\right )\right )+2 c^3 x^2 (g x-3 f)\right )+8 c^2 \left (a^3 (2 i+3 j x)-a^2 c \left (g+x^2 (3 i+4 j x)\right )-a c^2 x \left (3 f+h x^2\right )+2 c^3 f x^3\right )+b^4 \left (6 a j x-4 c j x^3\right )+3 b^5 j x^2\right )}{3 c^2 \left (4 a c+b^2\right )^2 (a+x (b-c x))^{3/2}}+\frac {i j \log \left (2 \sqrt {a+x (b-c x)}+\frac {i (b-2 c x)}{\sqrt {c}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x - c*x^2)^(5/2),x]

[Out]

(-2*(3*b^5*j*x^2 + b^4*(6*a*j*x - 4*c*j*x^3) + b^3*(3*a^2*j + 18*a*c*j*x^2 + c^2*(f + 3*g*x - x^2*(3*h + i*x))
) + 8*c^2*(2*c^3*f*x^3 + a^3*(2*i + 3*j*x) - a*c^2*x*(3*f + h*x^2) - a^2*c*(g + x^2*(3*i + 4*j*x))) + 4*b*c*(5
*a^3*j + 2*c^3*x^2*(-3*f + g*x) - 2*a^2*c*(h - 3*i*x) + 3*a*c^2*(f - x*(g - h*x + i*x^2))) + 2*b^2*c*(21*a^2*j
*x + c^2*x*(3*f + x*(-6*g + h*x)) + a*c*(g + x*(-6*h + 3*i*x - 14*j*x^2)))))/(3*c^2*(b^2 + 4*a*c)^2*(a + x*(b
- c*x))^(3/2)) + (I*j*Log[(I*(b - 2*c*x))/Sqrt[c] + 2*Sqrt[a + x*(b - c*x)]])/c^(5/2)

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fricas [B]  time = 95.78, size = 1385, normalized size = 3.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x^4+i*x^3+h*x^2+g*x+f)/(-c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*((b^4*c^2 + 8*a*b^2*c^3 + 16*a^2*c^4)*j*x^4 - 2*(b^5*c + 8*a*b^3*c^2 + 16*a^2*b*c^3)*j*x^3 + (b^6 + 6
*a*b^4*c - 32*a^3*c^3)*j*x^2 + 2*(a*b^5 + 8*a^2*b^3*c + 16*a^3*b*c^2)*j*x + (a^2*b^4 + 8*a^3*b^2*c + 16*a^4*c^
2)*j)*sqrt(-c)*log(8*c^2*x^2 - 8*b*c*x + b^2 - 4*sqrt(-c*x^2 + b*x + a)*(2*c*x - b)*sqrt(-c) - 4*a*c) - 4*(8*a
^2*b*c^3*h - 16*a^3*c^3*i - (16*c^6*f + 8*b*c^5*g + 2*(b^2*c^4 - 4*a*c^5)*h - (b^3*c^3 + 12*a*b*c^4)*i - 4*(b^
4*c^2 + 7*a*b^2*c^3 + 8*a^2*c^4)*j)*x^3 + 3*(8*b*c^5*f + 4*b^2*c^4*g + (b^3*c^3 - 4*a*b*c^4)*h - 2*(a*b^2*c^3
- 4*a^2*c^4)*i - (b^5*c + 6*a*b^3*c^2)*j)*x^2 - (b^3*c^3 + 12*a*b*c^4)*f - 2*(a*b^2*c^3 - 4*a^2*c^4)*g - (3*a^
2*b^3*c + 20*a^3*b*c^2)*j + 3*(4*a*b^2*c^3*h - 8*a^2*b*c^3*i - 2*(b^2*c^4 - 4*a*c^5)*f - (b^3*c^3 - 4*a*b*c^4)
*g - 2*(a*b^4*c + 7*a^2*b^2*c^2 + 4*a^3*c^3)*j)*x)*sqrt(-c*x^2 + b*x + a))/(a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a
^4*c^5 + (b^4*c^5 + 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 - 2*(b^5*c^4 + 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 +
6*a*b^4*c^4 - 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x), -1/3*(3*((b^4*c^2 + 8*a*b^2*c
^3 + 16*a^2*c^4)*j*x^4 - 2*(b^5*c + 8*a*b^3*c^2 + 16*a^2*b*c^3)*j*x^3 + (b^6 + 6*a*b^4*c - 32*a^3*c^3)*j*x^2 +
2*(a*b^5 + 8*a^2*b^3*c + 16*a^3*b*c^2)*j*x + (a^2*b^4 + 8*a^3*b^2*c + 16*a^4*c^2)*j)*sqrt(c)*arctan(1/2*sqrt(
-c*x^2 + b*x + a)*(2*c*x - b)*sqrt(c)/(c^2*x^2 - b*c*x - a*c)) - 2*(8*a^2*b*c^3*h - 16*a^3*c^3*i - (16*c^6*f +
8*b*c^5*g + 2*(b^2*c^4 - 4*a*c^5)*h - (b^3*c^3 + 12*a*b*c^4)*i - 4*(b^4*c^2 + 7*a*b^2*c^3 + 8*a^2*c^4)*j)*x^3
+ 3*(8*b*c^5*f + 4*b^2*c^4*g + (b^3*c^3 - 4*a*b*c^4)*h - 2*(a*b^2*c^3 - 4*a^2*c^4)*i - (b^5*c + 6*a*b^3*c^2)*
j)*x^2 - (b^3*c^3 + 12*a*b*c^4)*f - 2*(a*b^2*c^3 - 4*a^2*c^4)*g - (3*a^2*b^3*c + 20*a^3*b*c^2)*j + 3*(4*a*b^2*
c^3*h - 8*a^2*b*c^3*i - 2*(b^2*c^4 - 4*a*c^5)*f - (b^3*c^3 - 4*a*b*c^4)*g - 2*(a*b^4*c + 7*a^2*b^2*c^2 + 4*a^3
*c^3)*j)*x)*sqrt(-c*x^2 + b*x + a))/(a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 + 8*a*b^2*c^6 + 16*a^
2*c^7)*x^4 - 2*(b^5*c^4 + 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 + 6*a*b^4*c^4 - 32*a^3*c^6)*x^2 + 2*(a*b^
5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)]

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giac [A]  time = 0.45, size = 488, normalized size = 1.38 \[ -\frac {2 \, \sqrt {-c x^{2} + b x + a} {\left ({\left ({\left (\frac {{\left (16 \, c^{5} f + 8 \, b c^{4} g + 2 \, b^{2} c^{3} h - 8 \, a c^{4} h - b^{3} c^{2} i - 12 \, a b c^{3} i - 4 \, b^{4} c j - 28 \, a b^{2} c^{2} j - 32 \, a^{2} c^{3} j\right )} x}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} - \frac {3 \, {\left (8 \, b c^{4} f + 4 \, b^{2} c^{3} g + b^{3} c^{2} h - 4 \, a b c^{3} h - 2 \, a b^{2} c^{2} i + 8 \, a^{2} c^{3} i - b^{5} j - 6 \, a b^{3} c j\right )}}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c^{3} f - 8 \, a c^{4} f + b^{3} c^{2} g - 4 \, a b c^{3} g - 4 \, a b^{2} c^{2} h + 8 \, a^{2} b c^{2} i + 2 \, a b^{4} j + 14 \, a^{2} b^{2} c j + 8 \, a^{3} c^{2} j\right )}}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {b^{3} c^{2} f + 12 \, a b c^{3} f + 2 \, a b^{2} c^{2} g - 8 \, a^{2} c^{3} g - 8 \, a^{2} b c^{2} h + 16 \, a^{3} c^{2} i + 3 \, a^{2} b^{3} j + 20 \, a^{3} b c j}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} - b x - a\right )}^{2}} - \frac {j \log \left ({\left | 2 \, {\left (\sqrt {-c} x - \sqrt {-c x^{2} + b x + a}\right )} \sqrt {-c} + b \right |}\right )}{\sqrt {-c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x^4+i*x^3+h*x^2+g*x+f)/(-c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-2/3*sqrt(-c*x^2 + b*x + a)*((((16*c^5*f + 8*b*c^4*g + 2*b^2*c^3*h - 8*a*c^4*h - b^3*c^2*i - 12*a*b*c^3*i - 4*
b^4*c*j - 28*a*b^2*c^2*j - 32*a^2*c^3*j)*x/(b^4*c^2 + 8*a*b^2*c^3 + 16*a^2*c^4) - 3*(8*b*c^4*f + 4*b^2*c^3*g +
b^3*c^2*h - 4*a*b*c^3*h - 2*a*b^2*c^2*i + 8*a^2*c^3*i - b^5*j - 6*a*b^3*c*j)/(b^4*c^2 + 8*a*b^2*c^3 + 16*a^2*
c^4))*x + 3*(2*b^2*c^3*f - 8*a*c^4*f + b^3*c^2*g - 4*a*b*c^3*g - 4*a*b^2*c^2*h + 8*a^2*b*c^2*i + 2*a*b^4*j + 1
4*a^2*b^2*c*j + 8*a^3*c^2*j)/(b^4*c^2 + 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^3*c^2*f + 12*a*b*c^3*f + 2*a*b^2*c^2
*g - 8*a^2*c^3*g - 8*a^2*b*c^2*h + 16*a^3*c^2*i + 3*a^2*b^3*j + 20*a^3*b*c*j)/(b^4*c^2 + 8*a*b^2*c^3 + 16*a^2*
c^4))/(c*x^2 - b*x - a)^2 - j*log(abs(2*(sqrt(-c)*x - sqrt(-c*x^2 + b*x + a))*sqrt(-c) + b))/(sqrt(-c)*c^2)

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maple [B]  time = 0.02, size = 1453, normalized size = 4.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((j*x^4+i*x^3+h*x^2+g*x+f)/(-c*x^2+b*x+a)^(5/2),x)

[Out]

1/3*g/c/(-c*x^2+b*x+a)^(3/2)+j/c^(5/2)*arctan(c^(1/2)*(x-1/2/c*b)/(-c*x^2+b*x+a)^(1/2))+16/3*g*c*b/(-4*a*c-b^2
)^2/(-c*x^2+b*x+a)^(1/2)*x+1/12*i/c^2*b^3/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x-2/3*i/c*b^3/(-4*a*c-b^2)^2/(-c*x
^2+b*x+a)^(1/2)*x-1/2*i/c^2*b^2*a/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)-8*i*b*a/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2
)*x+4*i/c*b^2*a/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)-1/6*h/c*b^2/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x-1/3*h*a/c/
(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*b-16/3*h*a*c/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)*x+2*j/c^2*b^3*a/(-4*a*c-b^2
)^2/(-c*x^2+b*x+a)^(1/2)+j/c^2*b^2/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(1/2)*x+1/24*j/c^3*b^4/(-4*a*c-b^2)/(-c*x^2+b*x
+a)^(3/2)*x-1/3*j/c^2*b^4/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)*x-1/4*j/c^3*b^3*a/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3
/2)+1/2*h*x/c/(-c*x^2+b*x+a)^(3/2)+1/12*h/c^2*b/(-c*x^2+b*x+a)^(3/2)-8/3*g*b^2/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(
1/2)+2/3*f/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*b+i*x^2/c/(-c*x^2+b*x+a)^(3/2)-1/24*i/c^3*b^2/(-c*x^2+b*x+a)^(3/2
)-2/3*i*a/c^2/(-c*x^2+b*x+a)^(3/2)+1/3*j*x^3/c/(-c*x^2+b*x+a)^(3/2)-1/48*j/c^4*b^3/(-c*x^2+b*x+a)^(3/2)-j/c^2*
x/(-c*x^2+b*x+a)^(1/2)-1/2*j/c^3*b/(-c*x^2+b*x+a)^(1/2)+i/c*b*a/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x+1/2*j/c^2*
b^2*a/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x-4*j/c*b^2*a/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)*x-1/3*j/c^3*b*a/(-c*
x^2+b*x+a)^(3/2)-1/2*j/c^3*b^3/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(1/2)-1/4*i/c^2*b*x/(-c*x^2+b*x+a)^(3/2)-1/24*i/c^3
*b^4/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)+1/3*i/c^2*b^4/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)-4/3*f/(-4*a*c-b^2)/(-
c*x^2+b*x+a)^(3/2)*x*c+32/3*f*c^2/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)*x-16/3*f*c/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)
^(1/2)*b-2/3*g*b/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x+1/3*g/c*b^2/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)+1/12*h/c^2*
b^3/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)+4/3*h*b^2/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)*x-2/3*h/c*b^3/(-4*a*c-b^2)
^2/(-c*x^2+b*x+a)^(1/2)+2/3*h*a/(-4*a*c-b^2)/(-c*x^2+b*x+a)^(3/2)*x+8/3*h*a/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2
)*b+1/2*j/c^2*b*x^2/(-c*x^2+b*x+a)^(3/2)-1/8*j/c^3*b^2*x/(-c*x^2+b*x+a)^(3/2)-1/48*j/c^4*b^5/(-4*a*c-b^2)/(-c*
x^2+b*x+a)^(3/2)+1/6*j/c^3*b^5/(-4*a*c-b^2)^2/(-c*x^2+b*x+a)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, i {\left (\frac {32 \, a b x}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} - \frac {16 \, a b^{2}}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2} c} + \frac {b^{3} x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c^{2}} + \frac {2 \, {\left (b^{2} - 4 \, a c\right )} b x}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2} c} + \frac {6 \, a b x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c} - \frac {3 \, x^{2}}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} c} - \frac {{\left (b^{2} - 4 \, a c\right )} b^{2}}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2} c^{2}} - \frac {a b^{2}}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c^{2}} + \frac {2 \, a}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} c^{2}}\right )} + \frac {1}{3} \, g {\left (\frac {16 \, b c x}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} - \frac {8 \, b^{2}}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} + \frac {2 \, b x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )}} - \frac {b^{2}}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c} + \frac {1}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} c}\right )} + \frac {2}{3} \, f {\left (\frac {16 \, c^{2} x}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} - \frac {8 \, b c}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} + \frac {2 \, c x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )}} - \frac {b}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )}}\right )} + \frac {2}{3} \, h {\left (\frac {2 \, {\left (b^{2} - 4 \, a c\right )} x}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2}} + \frac {2 \, a x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )}} + \frac {b^{2} x}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c} - \frac {{\left (b^{2} - 4 \, a c\right )} b}{\sqrt {-c x^{2} + b x + a} {\left (b^{2} + 4 \, a c\right )}^{2} c} + \frac {a b}{{\left (-c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (b^{2} + 4 \, a c\right )} c}\right )} + j \int \frac {x^{4}}{{\left (c^{2} x^{4} - 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} - 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {-c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x^4+i*x^3+h*x^2+g*x+f)/(-c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

-1/3*i*(32*a*b*x/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2) - 16*a*b^2/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2*c
) + b^3*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)*c^2) + 2*(b^2 - 4*a*c)*b*x/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*
a*c)^2*c) + 6*a*b*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)*c) - 3*x^2/((-c*x^2 + b*x + a)^(3/2)*c) - (b^2 - 4
*a*c)*b^2/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2*c^2) - a*b^2/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)*c^2) +
2*a/((-c*x^2 + b*x + a)^(3/2)*c^2)) + 1/3*g*(16*b*c*x/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2) - 8*b^2/(sqrt(-
c*x^2 + b*x + a)*(b^2 + 4*a*c)^2) + 2*b*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)) - b^2/((-c*x^2 + b*x + a)^(
3/2)*(b^2 + 4*a*c)*c) + 1/((-c*x^2 + b*x + a)^(3/2)*c)) + 2/3*f*(16*c^2*x/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c
)^2) - 8*b*c/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2) + 2*c*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)) - b/((-
c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c))) + 2/3*h*(2*(b^2 - 4*a*c)*x/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2) + 2
*a*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)) + b^2*x/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)*c) - (b^2 - 4*a*
c)*b/(sqrt(-c*x^2 + b*x + a)*(b^2 + 4*a*c)^2*c) + a*b/((-c*x^2 + b*x + a)^(3/2)*(b^2 + 4*a*c)*c)) + j*integrat
e(x^4/((c^2*x^4 - 2*b*c*x^3 + 2*a*b*x + (b^2 - 2*a*c)*x^2 + a^2)*sqrt(-c*x^2 + b*x + a)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {j\,x^4+i\,x^3+h\,x^2+g\,x+f}{{\left (-c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x - c*x^2)^(5/2),x)

[Out]

int((f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x - c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x**4+i*x**3+h*x**2+g*x+f)/(-c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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