### 3.109 $$\int \frac{(g+h x)^2 (d+e x+f x^2)}{(a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=149 $\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (h^2 (2 c d-3 a f)+2 c g (2 e h+f g)\right )}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (4 (c d g-a (e h+2 f g))+h x (2 c d-3 a f))}{2 a c^2}-\frac{(g+h x)^2 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}}$

[Out]

-(((a*e - (c*d - a*f)*x)*(g + h*x)^2)/(a*c*Sqrt[a + c*x^2])) - (h*(4*(c*d*g - a*(2*f*g + e*h)) + (2*c*d - 3*a*
f)*h*x)*Sqrt[a + c*x^2])/(2*a*c^2) + (((2*c*d - 3*a*f)*h^2 + 2*c*g*(f*g + 2*e*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(2*c^(5/2))

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Rubi [A]  time = 0.183902, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.138, Rules used = {1645, 780, 217, 206} $\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (h^2 (2 c d-3 a f)+2 c g (2 e h+f g)\right )}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (4 (c d g-a (e h+2 f g))+h x (2 c d-3 a f))}{2 a c^2}-\frac{(g+h x)^2 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(((a*e - (c*d - a*f)*x)*(g + h*x)^2)/(a*c*Sqrt[a + c*x^2])) - (h*(4*(c*d*g - a*(2*f*g + e*h)) + (2*c*d - 3*a*
f)*h*x)*Sqrt[a + c*x^2])/(2*a*c^2) + (((2*c*d - 3*a*f)*h^2 + 2*c*g*(f*g + 2*e*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(2*c^(5/2))

Rule 1645

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

Rule 780

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

Rule 217

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

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

Mathematica [A]  time = 0.301613, size = 177, normalized size = 1.19 $\frac{\sqrt{c} \left (a^2 h (4 e h+8 f g+3 f h x)+a c \left (-2 d h (2 g+h x)-2 e \left (g^2+2 g h x-h^2 x^2\right )+f x \left (-2 g^2+4 g h x+h^2 x^2\right )\right )+2 c^2 d g^2 x\right )-a^{3/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 a f h^2-2 c \left (h (d h+2 e g)+f g^2\right )\right )}{2 a c^{5/2} \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[c]*(2*c^2*d*g^2*x + a^2*h*(8*f*g + 4*e*h + 3*f*h*x) + a*c*(-2*d*h*(2*g + h*x) - 2*e*(g^2 + 2*g*h*x - h^2
*x^2) + f*x*(-2*g^2 + 4*g*h*x + h^2*x^2))) - a^(3/2)*(3*a*f*h^2 - 2*c*(f*g^2 + h*(2*e*g + d*h)))*Sqrt[1 + (c*x
^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(2*a*c^(5/2)*Sqrt[a + c*x^2])

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Maple [B]  time = 0.058, size = 327, normalized size = 2.2 \begin{align*}{\frac{{h}^{2}f{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,af{h}^{2}x}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,af{h}^{2}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{h}^{2}{x}^{2}e}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{{x}^{2}ghf}{c\sqrt{c{x}^{2}+a}}}+2\,{\frac{a{h}^{2}e}{{c}^{2}\sqrt{c{x}^{2}+a}}}+4\,{\frac{aghf}{{c}^{2}\sqrt{c{x}^{2}+a}}}-{\frac{dx{h}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-2\,{\frac{egxh}{c\sqrt{c{x}^{2}+a}}}-{\frac{fx{g}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{d{h}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) egh}{{c}^{3/2}}}+{f{g}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{ghd}{c\sqrt{c{x}^{2}+a}}}-{\frac{e{g}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{g}^{2}dx}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.05239, size = 1142, normalized size = 7.66 \begin{align*} \left [-\frac{{\left (2 \, a^{2} c f g^{2} + 4 \, a^{2} c e g h +{\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} h^{2} +{\left (2 \, a c^{2} f g^{2} + 4 \, a c^{2} e g h +{\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} h^{2}\right )} x^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (a c^{2} f h^{2} x^{3} - 2 \, a c^{2} e g^{2} + 4 \, a^{2} c e h^{2} - 4 \,{\left (a c^{2} d - 2 \, a^{2} c f\right )} g h + 2 \,{\left (2 \, a c^{2} f g h + a c^{2} e h^{2}\right )} x^{2} -{\left (4 \, a c^{2} e g h - 2 \,{\left (c^{3} d - a c^{2} f\right )} g^{2} +{\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} h^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{4 \,{\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac{{\left (2 \, a^{2} c f g^{2} + 4 \, a^{2} c e g h +{\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} h^{2} +{\left (2 \, a c^{2} f g^{2} + 4 \, a c^{2} e g h +{\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} h^{2}\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (a c^{2} f h^{2} x^{3} - 2 \, a c^{2} e g^{2} + 4 \, a^{2} c e h^{2} - 4 \,{\left (a c^{2} d - 2 \, a^{2} c f\right )} g h + 2 \,{\left (2 \, a c^{2} f g h + a c^{2} e h^{2}\right )} x^{2} -{\left (4 \, a c^{2} e g h - 2 \,{\left (c^{3} d - a c^{2} f\right )} g^{2} +{\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} h^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((2*a^2*c*f*g^2 + 4*a^2*c*e*g*h + (2*a^2*c*d - 3*a^3*f)*h^2 + (2*a*c^2*f*g^2 + 4*a*c^2*e*g*h + (2*a*c^2*
d - 3*a^2*c*f)*h^2)*x^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(a*c^2*f*h^2*x^3 - 2*a*c^
2*e*g^2 + 4*a^2*c*e*h^2 - 4*(a*c^2*d - 2*a^2*c*f)*g*h + 2*(2*a*c^2*f*g*h + a*c^2*e*h^2)*x^2 - (4*a*c^2*e*g*h -
2*(c^3*d - a*c^2*f)*g^2 + (2*a*c^2*d - 3*a^2*c*f)*h^2)*x)*sqrt(c*x^2 + a))/(a*c^4*x^2 + a^2*c^3), -1/2*((2*a^
2*c*f*g^2 + 4*a^2*c*e*g*h + (2*a^2*c*d - 3*a^3*f)*h^2 + (2*a*c^2*f*g^2 + 4*a*c^2*e*g*h + (2*a*c^2*d - 3*a^2*c*
f)*h^2)*x^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (a*c^2*f*h^2*x^3 - 2*a*c^2*e*g^2 + 4*a^2*c*e*h^2 -
4*(a*c^2*d - 2*a^2*c*f)*g*h + 2*(2*a*c^2*f*g*h + a*c^2*e*h^2)*x^2 - (4*a*c^2*e*g*h - 2*(c^3*d - a*c^2*f)*g^2 +
(2*a*c^2*d - 3*a^2*c*f)*h^2)*x)*sqrt(c*x^2 + a))/(a*c^4*x^2 + a^2*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g + h x\right )^{2} \left (d + e x + f x^{2}\right )}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.188, size = 296, normalized size = 1.99 \begin{align*} \frac{{\left ({\left (\frac{f h^{2} x}{c} + \frac{2 \,{\left (2 \, a c^{3} f g h + a c^{3} h^{2} e\right )}}{a c^{4}}\right )} x + \frac{2 \, c^{4} d g^{2} - 2 \, a c^{3} f g^{2} - 2 \, a c^{3} d h^{2} + 3 \, a^{2} c^{2} f h^{2} - 4 \, a c^{3} g h e}{a c^{4}}\right )} x - \frac{2 \,{\left (2 \, a c^{3} d g h - 4 \, a^{2} c^{2} f g h + a c^{3} g^{2} e - 2 \, a^{2} c^{2} h^{2} e\right )}}{a c^{4}}}{2 \, \sqrt{c x^{2} + a}} - \frac{{\left (2 \, c f g^{2} + 2 \, c d h^{2} - 3 \, a f h^{2} + 4 \, c g h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((f*h^2*x/c + 2*(2*a*c^3*f*g*h + a*c^3*h^2*e)/(a*c^4))*x + (2*c^4*d*g^2 - 2*a*c^3*f*g^2 - 2*a*c^3*d*h^2 +
3*a^2*c^2*f*h^2 - 4*a*c^3*g*h*e)/(a*c^4))*x - 2*(2*a*c^3*d*g*h - 4*a^2*c^2*f*g*h + a*c^3*g^2*e - 2*a^2*c^2*h^
2*e)/(a*c^4))/sqrt(c*x^2 + a) - 1/2*(2*c*f*g^2 + 2*c*d*h^2 - 3*a*f*h^2 + 4*c*g*h*e)*log(abs(-sqrt(c)*x + sqrt(
c*x^2 + a)))/c^(5/2)