### 3.563 $$\int \frac{(d+e x)^2}{\sqrt{a+c x^2}} \, dx$$

Optimal. Leaf size=86 $\frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0402104, antiderivative size = 86, 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 = {743, 641, 217, 206} $\frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2/Sqrt[a + c*x^2],x]

[Out]

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

Rule 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && 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}{\sqrt{a+c x^2}} \, dx &=\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\int \frac{2 c d^2-a e^2+3 c d e x}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0424227, size = 71, normalized size = 0.83 $\frac{\left (2 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} e \sqrt{a+c x^2} (4 d+e x)}{2 c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[c]*e*(4*d + e*x)*Sqrt[a + c*x^2] + (2*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2*c^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 84, normalized size = 1. \begin{align*}{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{a{e}^{2}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+a}}{c}}+{{d}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/(c*x^2+a)^(1/2),x)

[Out]

1/2*e^2*x/c*(c*x^2+a)^(1/2)-1/2*e^2*a/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+2*d*e*(c*x^2+a)^(1/2)/c+d^2*ln(x*c
^(1/2)+(c*x^2+a)^(1/2))/c^(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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.92376, size = 327, normalized size = 3.8 \begin{align*} \left [-\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (c e^{2} x + 4 \, c d e\right )} \sqrt{c x^{2} + a}}{4 \, c^{2}}, -\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (c e^{2} x + 4 \, c d e\right )} \sqrt{c 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)^(1/2),x, algorithm="fricas")

[Out]

[-1/4*((2*c*d^2 - a*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(c*e^2*x + 4*c*d*e)*sqrt(
c*x^2 + a))/c^2, -1/2*((2*c*d^2 - a*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (c*e^2*x + 4*c*d*e)*sqr
t(c*x^2 + a))/c^2]

________________________________________________________________________________________

Sympy [A]  time = 4.90411, size = 158, normalized size = 1.84 \begin{align*} \frac{\sqrt{a} e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{a e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{2} \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + 2 d e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

sqrt(a)*e**2*x*sqrt(1 + c*x**2/a)/(2*c) - a*e**2*asinh(sqrt(c)*x/sqrt(a))/(2*c**(3/2)) + d**2*Piecewise((sqrt(
-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0) & (c < 0)), (sqrt(a/c)*asinh(x*sqrt(c/a))/sqrt(a), (a > 0) & (c > 0)
), (sqrt(-a/c)*acosh(x*sqrt(-c/a))/sqrt(-a), (c > 0) & (a < 0))) + 2*d*e*Piecewise((x**2/(2*sqrt(a)), Eq(c, 0)
), (sqrt(a + c*x**2)/c, True))

________________________________________________________________________________________

Giac [A]  time = 1.37338, size = 85, normalized size = 0.99 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (\frac{x e^{2}}{c} + \frac{4 \, d e}{c}\right )} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, 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)^(1/2),x, algorithm="giac")

[Out]

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