### 3.571 $$\int \frac{(d+e x)^2}{(a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=83 $\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}}$

[Out]

-(((a*e - c*d*x)*(d + e*x))/(a*c*Sqrt[a + c*x^2])) - (d*e*Sqrt[a + c*x^2])/(a*c) + (e^2*ArcTanh[(Sqrt[c]*x)/Sq
rt[a + c*x^2]])/c^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.0345132, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.21, Rules used = {739, 641, 217, 206} $\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2/(a + c*x^2)^(3/2),x]

[Out]

-(((a*e - c*d*x)*(d + e*x))/(a*c*Sqrt[a + c*x^2])) - (d*e*Sqrt[a + c*x^2])/(a*c) + (e^2*ArcTanh[(Sqrt[c]*x)/Sq
rt[a + c*x^2]])/c^(3/2)

Rule 739

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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{(d+e x)^2}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}+\frac{\int \frac{a e^2-c d e x}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0697005, size = 69, normalized size = 0.83 $\frac{e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{a c \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*a*d*e + c*d^2*x - a*e^2*x)/(a*c*Sqrt[a + c*x^2]) + (e^2*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(3/2)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 76, normalized size = 0.9 \begin{align*} -{\frac{{e}^{2}x}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{{e}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{de}{c\sqrt{c{x}^{2}+a}}}+{\frac{{d}^{2}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^2*x/c/(c*x^2+a)^(1/2)+e^2/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))-2*d*e/c/(c*x^2+a)^(1/2)+d^2*x/a/(c*x^2+a)^(
1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.99505, size = 428, normalized size = 5.16 \begin{align*} \left [\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (2 \, a c d e -{\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, a c d e -{\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{a c^{3} x^{2} + a^{2} c^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a*c*e^2*x^2 + a^2*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(2*a*c*d*e - (c^2*d^
2 - a*c*e^2)*x)*sqrt(c*x^2 + a))/(a*c^3*x^2 + a^2*c^2), -((a*c*e^2*x^2 + a^2*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/s
qrt(c*x^2 + a)) + (2*a*c*d*e - (c^2*d^2 - a*c*e^2)*x)*sqrt(c*x^2 + a))/(a*c^3*x^2 + a^2*c^2)]

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.39254, size = 93, normalized size = 1.12 \begin{align*} -\frac{\frac{2 \, d e}{c} - \frac{{\left (c^{2} d^{2} - a c e^{2}\right )} x}{a c^{2}}}{\sqrt{c x^{2} + a}} - \frac{e^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*d*e/c - (c^2*d^2 - a*c*e^2)*x/(a*c^2))/sqrt(c*x^2 + a) - e^2*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2
)