### 3.114 $$\int \frac{d+e x+f x^2}{(g+h x)^3 (a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=374 $\frac{c x \left (a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )+c^2 d g^3\right )+a \left (a^2 f h^3-a c h \left (3 f g^2-h (3 e g-d h)\right )-c^2 g^2 (e g-3 d h)\right )}{a \sqrt{a+c x^2} \left (a h^2+c g^2\right )^3}-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4-a c h^2 \left (3 d h^2-9 e g h+11 f g^2\right )+2 c^2 g^2 \left (6 d h^2-3 e g h+f g^2\right )\right )}{2 \left (a h^2+c g^2\right )^{7/2}}-\frac{h \sqrt{a+c x^2} \left (d h^2-e g h+f g^2\right )}{2 (g+h x)^2 \left (a h^2+c g^2\right )^2}+\frac{h \sqrt{a+c x^2} \left (2 a h^2 (2 f g-e h)-c g \left (3 f g^2-h (5 e g-7 d h)\right )\right )}{2 (g+h x) \left (a h^2+c g^2\right )^3}$

[Out]

(a*(a^2*f*h^3 - c^2*g^2*(e*g - 3*d*h) - a*c*h*(3*f*g^2 - h*(3*e*g - d*h))) + c*(c^2*d*g^3 + a^2*h^2*(3*f*g - e
*h) - a*c*g*(f*g^2 - 3*h*(e*g - d*h)))*x)/(a*(c*g^2 + a*h^2)^3*Sqrt[a + c*x^2]) - (h*(f*g^2 - e*g*h + d*h^2)*S
qrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^2*(g + h*x)^2) + (h*(2*a*h^2*(2*f*g - e*h) - c*g*(3*f*g^2 - h*(5*e*g - 7*d*
h)))*Sqrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^3*(g + h*x)) - ((2*a^2*f*h^4 - a*c*h^2*(11*f*g^2 - 9*e*g*h + 3*d*h^2)
+ 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*(c*
g^2 + a*h^2)^(7/2))

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Rubi [A]  time = 1.02786, antiderivative size = 372, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.172, Rules used = {1647, 1651, 807, 725, 206} $\frac{c x \left (a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )+c^2 d g^3\right )+a \left (a^2 f h^3-a c h \left (3 f g^2-h (3 e g-d h)\right )-c^2 g^2 (e g-3 d h)\right )}{a \sqrt{a+c x^2} \left (a h^2+c g^2\right )^3}-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4-a c h^2 \left (3 d h^2-9 e g h+11 f g^2\right )+2 c^2 g^2 \left (6 d h^2-3 e g h+f g^2\right )\right )}{2 \left (a h^2+c g^2\right )^{7/2}}-\frac{h \sqrt{a+c x^2} \left (d h^2-e g h+f g^2\right )}{2 (g+h x)^2 \left (a h^2+c g^2\right )^2}-\frac{h \sqrt{a+c x^2} \left (-2 a h^2 (2 f g-e h)-c g h (5 e g-7 d h)+3 c f g^3\right )}{2 (g+h x) \left (a h^2+c g^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*(a^2*f*h^3 - c^2*g^2*(e*g - 3*d*h) - a*c*h*(3*f*g^2 - h*(3*e*g - d*h))) + c*(c^2*d*g^3 + a^2*h^2*(3*f*g - e
*h) - a*c*g*(f*g^2 - 3*h*(e*g - d*h)))*x)/(a*(c*g^2 + a*h^2)^3*Sqrt[a + c*x^2]) - (h*(f*g^2 - e*g*h + d*h^2)*S
qrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^2*(g + h*x)^2) - (h*(3*c*f*g^3 - c*g*h*(5*e*g - 7*d*h) - 2*a*h^2*(2*f*g - e
*h))*Sqrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^3*(g + h*x)) - ((2*a^2*f*h^4 - a*c*h^2*(11*f*g^2 - 9*e*g*h + 3*d*h^2)
+ 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*(c*
g^2 + a*h^2)^(7/2))

Rule 1647

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx &=\frac{a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt{a+c x^2}}-\frac{\int \frac{-\frac{a c \left (a^2 d h^6+c^2 g^4 \left (f g^2-3 e g h+6 d h^2\right )-a c g^2 h^2 \left (3 f g^2-h (e g+3 d h)\right )\right )}{\left (c g^2+a h^2\right )^3}-\frac{a c h^2 \left (a^2 e h^4-c^2 g^3 (3 e g-8 d h)-2 a c g^2 h (4 f g-3 e h)\right ) x}{\left (c g^2+a h^2\right )^3}-\frac{a c h^3 \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right ) x^2}{\left (c g^2+a h^2\right )^3}}{(g+h x)^3 \sqrt{a+c x^2}} \, dx}{a c}\\ &=\frac{a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt{a+c x^2}}-\frac{h \left (f g^2-e g h+d h^2\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}+\frac{\int \frac{-\frac{2 a c \left (a^2 h^4 (f g-e h)-c^2 g^3 \left (f g^2-3 e g h+6 d h^2\right )+2 a c g h^2 \left (2 f g^2-h (e g+d h)\right )\right )}{\left (c g^2+a h^2\right )^2}+\frac{a c h \left (2 a^2 f h^4-c^2 \left (f g^4+g^2 h (e g-5 d h)\right )-a c h^2 \left (7 f g^2-h (7 e g-3 d h)\right )\right ) x}{\left (c g^2+a h^2\right )^2}}{(g+h x)^2 \sqrt{a+c x^2}} \, dx}{2 a c \left (c g^2+a h^2\right )}\\ &=\frac{a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt{a+c x^2}}-\frac{h \left (f g^2-e g h+d h^2\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}+\frac{\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )^3}\\ &=\frac{a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt{a+c x^2}}-\frac{h \left (f g^2-e g h+d h^2\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}-\frac{\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c g^2+a h^2-x^2} \, dx,x,\frac{a h-c g x}{\sqrt{a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^3}\\ &=\frac{a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt{a+c x^2}}-\frac{h \left (f g^2-e g h+d h^2\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac{h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt{a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}-\frac{\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.27153, size = 404, normalized size = 1.08 $\frac{1}{2} \left (-\frac{\sqrt{a+c x^2} \left (\frac{2 \left (a^2 c h (h (d h-3 e g+e h x)+3 f g (g-h x))-a^3 f h^3+a c^2 g \left (3 d h (h x-g)+e g (g-3 h x)+f g^2 x\right )-c^3 d g^3 x\right )}{a \left (a+c x^2\right )}+\frac{h \left (a h^2+c g^2\right ) \left (h (d h-e g)+f g^2\right )}{(g+h x)^2}+\frac{h \left (2 a h^2 (e h-2 f g)+c g h (7 d h-5 e g)+3 c f g^3\right )}{g+h x}\right )}{\left (a h^2+c g^2\right )^3}-\frac{\log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (2 a^2 f h^4+a c h^2 \left (-3 d h^2+9 e g h-11 f g^2\right )+2 c^2 g^2 \left (6 d h^2-3 e g h+f g^2\right )\right )}{\left (a h^2+c g^2\right )^{7/2}}+\frac{\log (g+h x) \left (2 a^2 f h^4+a c h^2 \left (-3 d h^2+9 e g h-11 f g^2\right )+2 c^2 g^2 \left (6 d h^2-3 e g h+f g^2\right )\right )}{\left (a h^2+c g^2\right )^{7/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-((Sqrt[a + c*x^2]*((h*(c*g^2 + a*h^2)*(f*g^2 + h*(-(e*g) + d*h)))/(g + h*x)^2 + (h*(3*c*f*g^3 + c*g*h*(-5*e*
g + 7*d*h) + 2*a*h^2*(-2*f*g + e*h)))/(g + h*x) + (2*(-(a^3*f*h^3) - c^3*d*g^3*x + a*c^2*g*(f*g^2*x + e*g*(g -
3*h*x) + 3*d*h*(-g + h*x)) + a^2*c*h*(3*f*g*(g - h*x) + h*(-3*e*g + d*h + e*h*x))))/(a*(a + c*x^2))))/(c*g^2
+ a*h^2)^3) + ((2*a^2*f*h^4 + a*c*h^2*(-11*f*g^2 + 9*e*g*h - 3*d*h^2) + 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))
*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) - ((2*a^2*f*h^4 + a*c*h^2*(-11*f*g^2 + 9*e*g*h - 3*d*h^2) + 2*c^2*g^2*(f*
g^2 - 3*e*g*h + 6*d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/2

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Maple [B]  time = 0.258, size = 2584, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/h^2/(a*h^2+c*g^2)/(x+g/h)/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*f*g+1/2/h^2/(a*h^2+c*g^2)/(x
+g/h)^2/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*e*g-1/2/h^3/(a*h^2+c*g^2)/(x+g/h)^2/((x+g/h)^2*c
-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*f*g^2-5/2*c*g/(a*h^2+c*g^2)^2/(x+g/h)/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(
a*h^2+c*g^2)/h^2)^(1/2)*d+15/2*h*c^2*g^2/(a*h^2+c*g^2)^3/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)
*d+15/2/h*c^2*g^4/(a*h^2+c*g^2)^3/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*f+15/2*c^2*g^3/(a*h^2+
c*g^2)^3/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h
)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*e-15/2/h*c/(a*h^2+c*g^2)^2/((x+g/h)^2*c-2*c*g/h*(x+g/
h)+(a*h^2+c*g^2)/h^2)^(1/2)*f*g^2+3/2*h*c/(a*h^2+c*g^2)^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*
c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*d-9/
2*c/(a*h^2+c*g^2)^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1
/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*e*g-15/2*c^2*g^3/(a*h^2+c*g^2)^3/((x+g/h)^
2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*e+19/2/h*c^2*g^2/(a*h^2+c*g^2)^2/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(
a*h^2+c*g^2)/h^2)^(1/2)*x*e-15/2/h*c^3*g^4/(a*h^2+c*g^2)^3/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(
1/2)*x*e+15/2/h^2*c^3*g^5/(a*h^2+c*g^2)^3/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*f+5*f/h^2*
g/(a*h^2+c*g^2)/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*c-25/2/h^2*c^2*g^3/(a*h^2+c*g^2)^2/a
/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*f+9/2*c/(a*h^2+c*g^2)^2/((x+g/h)^2*c-2*c*g/h*(x+g/h)+
(a*h^2+c*g^2)/h^2)^(1/2)*e*g-f/h/(a*h^2+c*g^2)/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/
h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))-1/2/h/(a*h^2+c*
g^2)/(x+g/h)^2/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*d-1/h/(a*h^2+c*g^2)/(x+g/h)/((x+g/h)^2*c-
2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*e-3/2*h*c/(a*h^2+c*g^2)^2/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/
h^2)^(1/2)*d+f/h/(a*h^2+c*g^2)/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)-2/h/(a*h^2+c*g^2)/a/((x+g
/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*c*e+15/2/h*c/(a*h^2+c*g^2)^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((
2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2
)^(1/2))/(x+g/h))*f*g^2+5/2/h*c*g^2/(a*h^2+c*g^2)^2/(x+g/h)/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1
/2)*e-5/2/h^2*c*g^3/(a*h^2+c*g^2)^2/(x+g/h)/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*f+15/2*c^3*g
^3/(a*h^2+c*g^2)^3/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*d-15/2*h*c^2*g^2/(a*h^2+c*g^2)^3/
((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c
*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*d-15/2/h*c^2*g^4/(a*h^2+c*g^2)^3/((a*h^2+c*g^2)/h^2)^(1/2)*ln(
(2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+g/h)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^
2)^(1/2))/(x+g/h))*f-13/2*c^2*g/(a*h^2+c*g^2)^2/a/((x+g/h)^2*c-2*c*g/h*(x+g/h)+(a*h^2+c*g^2)/h^2)^(1/2)*x*d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 110.411, size = 5783, normalized size = 15.46 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((2*a^2*c^2*f*g^6 - 6*a^2*c^2*e*g^5*h + 9*a^3*c*e*g^3*h^3 + (12*a^2*c^2*d - 11*a^3*c*f)*g^4*h^2 - (3*a^3*
c*d - 2*a^4*f)*g^2*h^4 + (2*a*c^3*f*g^4*h^2 - 6*a*c^3*e*g^3*h^3 + 9*a^2*c^2*e*g*h^5 + (12*a*c^3*d - 11*a^2*c^2
*f)*g^2*h^4 - (3*a^2*c^2*d - 2*a^3*c*f)*h^6)*x^4 + 2*(2*a*c^3*f*g^5*h - 6*a*c^3*e*g^4*h^2 + 9*a^2*c^2*e*g^2*h^
4 + (12*a*c^3*d - 11*a^2*c^2*f)*g^3*h^3 - (3*a^2*c^2*d - 2*a^3*c*f)*g*h^5)*x^3 + (2*a*c^3*f*g^6 - 6*a*c^3*e*g^
5*h + 3*a^2*c^2*e*g^3*h^3 + 9*a^3*c*e*g*h^5 + 3*(4*a*c^3*d - 3*a^2*c^2*f)*g^4*h^2 + 9*(a^2*c^2*d - a^3*c*f)*g^
2*h^4 - (3*a^3*c*d - 2*a^4*f)*h^6)*x^2 + 2*(2*a^2*c^2*f*g^5*h - 6*a^2*c^2*e*g^4*h^2 + 9*a^3*c*e*g^2*h^4 + (12*
a^2*c^2*d - 11*a^3*c*f)*g^3*h^3 - (3*a^3*c*d - 2*a^4*f)*g*h^5)*x)*sqrt(c*g^2 + a*h^2)*log((2*a*c*g*h*x - a*c*g
^2 - 2*a^2*h^2 - (2*c^2*g^2 + a*c*h^2)*x^2 - 2*sqrt(c*g^2 + a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a))/(h^2*x^2 + 2
*g*h*x + g^2)) - 2*(2*a*c^3*e*g^7 - 10*a^2*c^2*e*g^5*h^2 - 11*a^3*c*e*g^3*h^4 + a^4*e*g*h^6 + a^4*d*h^7 - 2*(3
*a*c^3*d - 5*a^2*c^2*f)*g^6*h + (4*a^2*c^2*d + 5*a^3*c*f)*g^4*h^3 + (11*a^3*c*d - 5*a^4*f)*g^2*h^5 - (11*a*c^3
*e*g^4*h^3 + 7*a^2*c^2*e*g^2*h^5 - 4*a^3*c*e*h^7 + (2*c^4*d - 5*a*c^3*f)*g^5*h^2 - (11*a*c^3*d - 5*a^2*c^2*f)*
g^3*h^4 - (13*a^2*c^2*d - 10*a^3*c*f)*g*h^6)*x^3 - (16*a*c^3*e*g^5*h^2 + 17*a^2*c^2*e*g^3*h^4 + a^3*c*e*g*h^6
+ 4*(c^4*d - 2*a*c^3*f)*g^6*h - (10*a*c^3*d - a^2*c^2*f)*g^4*h^3 - (17*a^2*c^2*d - 11*a^3*c*f)*g^2*h^5 - (3*a^
3*c*d - 2*a^4*f)*h^7)*x^2 - (2*a*c^3*e*g^6*h + 17*a^2*c^2*e*g^4*h^3 + 13*a^3*c*e*g^2*h^5 - 2*a^4*e*h^7 + 2*(c^
4*d - a*c^3*f)*g^7 + (8*a*c^3*d - 11*a^2*c^2*f)*g^5*h^2 - (5*a^2*c^2*d + a^3*c*f)*g^3*h^4 - (11*a^3*c*d - 8*a^
4*f)*g*h^6)*x)*sqrt(c*x^2 + a))/(a^2*c^4*g^10 + 4*a^3*c^3*g^8*h^2 + 6*a^4*c^2*g^6*h^4 + 4*a^5*c*g^4*h^6 + a^6*
g^2*h^8 + (a*c^5*g^8*h^2 + 4*a^2*c^4*g^6*h^4 + 6*a^3*c^3*g^4*h^6 + 4*a^4*c^2*g^2*h^8 + a^5*c*h^10)*x^4 + 2*(a*
c^5*g^9*h + 4*a^2*c^4*g^7*h^3 + 6*a^3*c^3*g^5*h^5 + 4*a^4*c^2*g^3*h^7 + a^5*c*g*h^9)*x^3 + (a*c^5*g^10 + 5*a^2
*c^4*g^8*h^2 + 10*a^3*c^3*g^6*h^4 + 10*a^4*c^2*g^4*h^6 + 5*a^5*c*g^2*h^8 + a^6*h^10)*x^2 + 2*(a^2*c^4*g^9*h +
4*a^3*c^3*g^7*h^3 + 6*a^4*c^2*g^5*h^5 + 4*a^5*c*g^3*h^7 + a^6*g*h^9)*x), -1/2*((2*a^2*c^2*f*g^6 - 6*a^2*c^2*e*
g^5*h + 9*a^3*c*e*g^3*h^3 + (12*a^2*c^2*d - 11*a^3*c*f)*g^4*h^2 - (3*a^3*c*d - 2*a^4*f)*g^2*h^4 + (2*a*c^3*f*g
^4*h^2 - 6*a*c^3*e*g^3*h^3 + 9*a^2*c^2*e*g*h^5 + (12*a*c^3*d - 11*a^2*c^2*f)*g^2*h^4 - (3*a^2*c^2*d - 2*a^3*c*
f)*h^6)*x^4 + 2*(2*a*c^3*f*g^5*h - 6*a*c^3*e*g^4*h^2 + 9*a^2*c^2*e*g^2*h^4 + (12*a*c^3*d - 11*a^2*c^2*f)*g^3*h
^3 - (3*a^2*c^2*d - 2*a^3*c*f)*g*h^5)*x^3 + (2*a*c^3*f*g^6 - 6*a*c^3*e*g^5*h + 3*a^2*c^2*e*g^3*h^3 + 9*a^3*c*e
*g*h^5 + 3*(4*a*c^3*d - 3*a^2*c^2*f)*g^4*h^2 + 9*(a^2*c^2*d - a^3*c*f)*g^2*h^4 - (3*a^3*c*d - 2*a^4*f)*h^6)*x^
2 + 2*(2*a^2*c^2*f*g^5*h - 6*a^2*c^2*e*g^4*h^2 + 9*a^3*c*e*g^2*h^4 + (12*a^2*c^2*d - 11*a^3*c*f)*g^3*h^3 - (3*
a^3*c*d - 2*a^4*f)*g*h^5)*x)*sqrt(-c*g^2 - a*h^2)*arctan(sqrt(-c*g^2 - a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a)/(a
*c*g^2 + a^2*h^2 + (c^2*g^2 + a*c*h^2)*x^2)) + (2*a*c^3*e*g^7 - 10*a^2*c^2*e*g^5*h^2 - 11*a^3*c*e*g^3*h^4 + a^
4*e*g*h^6 + a^4*d*h^7 - 2*(3*a*c^3*d - 5*a^2*c^2*f)*g^6*h + (4*a^2*c^2*d + 5*a^3*c*f)*g^4*h^3 + (11*a^3*c*d -
5*a^4*f)*g^2*h^5 - (11*a*c^3*e*g^4*h^3 + 7*a^2*c^2*e*g^2*h^5 - 4*a^3*c*e*h^7 + (2*c^4*d - 5*a*c^3*f)*g^5*h^2 -
(11*a*c^3*d - 5*a^2*c^2*f)*g^3*h^4 - (13*a^2*c^2*d - 10*a^3*c*f)*g*h^6)*x^3 - (16*a*c^3*e*g^5*h^2 + 17*a^2*c^
2*e*g^3*h^4 + a^3*c*e*g*h^6 + 4*(c^4*d - 2*a*c^3*f)*g^6*h - (10*a*c^3*d - a^2*c^2*f)*g^4*h^3 - (17*a^2*c^2*d -
11*a^3*c*f)*g^2*h^5 - (3*a^3*c*d - 2*a^4*f)*h^7)*x^2 - (2*a*c^3*e*g^6*h + 17*a^2*c^2*e*g^4*h^3 + 13*a^3*c*e*g
^2*h^5 - 2*a^4*e*h^7 + 2*(c^4*d - a*c^3*f)*g^7 + (8*a*c^3*d - 11*a^2*c^2*f)*g^5*h^2 - (5*a^2*c^2*d + a^3*c*f)*
g^3*h^4 - (11*a^3*c*d - 8*a^4*f)*g*h^6)*x)*sqrt(c*x^2 + a))/(a^2*c^4*g^10 + 4*a^3*c^3*g^8*h^2 + 6*a^4*c^2*g^6*
h^4 + 4*a^5*c*g^4*h^6 + a^6*g^2*h^8 + (a*c^5*g^8*h^2 + 4*a^2*c^4*g^6*h^4 + 6*a^3*c^3*g^4*h^6 + 4*a^4*c^2*g^2*h
^8 + a^5*c*h^10)*x^4 + 2*(a*c^5*g^9*h + 4*a^2*c^4*g^7*h^3 + 6*a^3*c^3*g^5*h^5 + 4*a^4*c^2*g^3*h^7 + a^5*c*g*h^
9)*x^3 + (a*c^5*g^10 + 5*a^2*c^4*g^8*h^2 + 10*a^3*c^3*g^6*h^4 + 10*a^4*c^2*g^4*h^6 + 5*a^5*c*g^2*h^8 + a^6*h^1
0)*x^2 + 2*(a^2*c^4*g^9*h + 4*a^3*c^3*g^7*h^3 + 6*a^4*c^2*g^5*h^5 + 4*a^5*c*g^3*h^7 + a^6*g*h^9)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)/(h*x+g)**3/(c*x**2+a)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.38954, size = 1944, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

((c^6*d*g^9 - a*c^5*f*g^9 - 6*a^2*c^4*d*g^5*h^4 + 6*a^3*c^3*f*g^5*h^4 - 8*a^3*c^3*d*g^3*h^6 + 8*a^4*c^2*f*g^3*
h^6 - 3*a^4*c^2*d*g*h^8 + 3*a^5*c*f*g*h^8 + 3*a*c^5*g^8*h*e + 8*a^2*c^4*g^6*h^3*e + 6*a^3*c^3*g^4*h^5*e - a^5*
c*h^9*e)*x/(a*c^6*g^12 + 6*a^2*c^5*g^10*h^2 + 15*a^3*c^4*g^8*h^4 + 20*a^4*c^3*g^6*h^6 + 15*a^5*c^2*g^4*h^8 + 6
*a^6*c*g^2*h^10 + a^7*h^12) + (3*a*c^5*d*g^8*h - 3*a^2*c^4*f*g^8*h + 8*a^2*c^4*d*g^6*h^3 - 8*a^3*c^3*f*g^6*h^3
+ 6*a^3*c^3*d*g^4*h^5 - 6*a^4*c^2*f*g^4*h^5 - a^5*c*d*h^9 + a^6*f*h^9 - a*c^5*g^9*e + 6*a^3*c^3*g^5*h^4*e + 8
*a^4*c^2*g^3*h^6*e + 3*a^5*c*g*h^8*e)/(a*c^6*g^12 + 6*a^2*c^5*g^10*h^2 + 15*a^3*c^4*g^8*h^4 + 20*a^4*c^3*g^6*h
^6 + 15*a^5*c^2*g^4*h^8 + 6*a^6*c*g^2*h^10 + a^7*h^12))/sqrt(c*x^2 + a) - (2*c^2*f*g^4 + 12*c^2*d*g^2*h^2 - 11
*a*c*f*g^2*h^2 - 3*a*c*d*h^4 + 2*a^2*f*h^4 - 6*c^2*g^3*h*e + 9*a*c*g*h^3*e)*arctan(((sqrt(c)*x - sqrt(c*x^2 +
a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^3*g^6 + 3*a*c^2*g^4*h^2 + 3*a^2*c*g^2*h^4 + a^3*h^6)*sqrt(-c*g^2
- a*h^2)) - (2*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*f*g^4*h + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d*g^2*h^3 -
5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*f*g^2*h^3 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*d*h^5 - 4*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*c^2*g^3*h^2*e + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*g*h^4*e + 6*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*c^(5/2)*f*g^5 + 14*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*d*g^3*h^2 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))
^2*a*c^(3/2)*f*g^3*h^2 - 7*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*g*h^4 + 4*(sqrt(c)*x - sqrt(c*x^2 + a))
^2*a^2*sqrt(c)*f*g*h^4 - 10*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*g^4*h*e + 9*(sqrt(c)*x - sqrt(c*x^2 + a))^
2*a*c^(3/2)*g^2*h^3*e - 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*sqrt(c)*h^5*e - 10*(sqrt(c)*x - sqrt(c*x^2 + a))
*a*c^2*f*g^4*h - 22*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*d*g^2*h^3 + 11*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*f*g
^2*h^3 - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*d*h^5 + 16*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*g^3*h^2*e - 5*(sqr
t(c)*x - sqrt(c*x^2 + a))*a^2*c*g*h^4*e + 3*a^2*c^(3/2)*f*g^3*h^2 + 7*a^2*c^(3/2)*d*g*h^4 - 4*a^3*sqrt(c)*f*g*
h^4 - 5*a^2*c^(3/2)*g^2*h^3*e + 2*a^3*sqrt(c)*h^5*e)/((c^3*g^6 + 3*a*c^2*g^4*h^2 + 3*a^2*c*g^2*h^4 + a^3*h^6)*
((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^2)