### 3.562 $$\int \frac{(d+e x)^3}{\sqrt{a+c x^2}} \, dx$$

Optimal. Leaf size=110 $\frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c}$

[Out]

(e*(d + e*x)^2*Sqrt[a + c*x^2])/(3*c) + (e*(4*(4*c*d^2 - a*e^2) + 5*c*d*e*x)*Sqrt[a + c*x^2])/(6*c^2) + (d*(2*
c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0753626, antiderivative size = 110, 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, 780, 217, 206} $\frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/Sqrt[a + c*x^2],x]

[Out]

(e*(d + e*x)^2*Sqrt[a + c*x^2])/(3*c) + (e*(4*(4*c*d^2 - a*e^2) + 5*c*d*e*x)*Sqrt[a + c*x^2])/(6*c^2) + (d*(2*
c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[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 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{(d+e x)^3}{\sqrt{a+c x^2}} \, dx &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{\int \frac{(d+e x) \left (3 c d^2-2 a e^2+5 c d e x\right )}{\sqrt{a+c x^2}} \, dx}{3 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{\left (d \left (2 c d^2-3 a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{\left (d \left (2 c d^2-3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{d \left (2 c d^2-3 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 = 0.0623585, size = 92, normalized size = 0.84 $\frac{e \sqrt{a+c x^2} \left (c \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 a e^2\right )+3 \sqrt{c} d \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*Sqrt[c]*d*(2*c*d^2 - 3*a*e^2)*Log[c*x + S
qrt[c]*Sqrt[a + c*x^2]])/(6*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 126, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+a}}-{\frac{2\,a{e}^{3}}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,ad{e}^{2}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+a}}{c}}+{{d}^{3}\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)^3/(c*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.90701, size = 428, normalized size = 3.89 \begin{align*} \left [-\frac{3 \,{\left (2 \, c d^{3} - 3 \, a d 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 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{12 \, c^{2}}, -\frac{3 \,{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{6 \, c^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(2*c*d^3 - 3*a*d*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(2*c*e^3*x^2 + 9*c
*d*e^2*x + 18*c*d^2*e - 4*a*e^3)*sqrt(c*x^2 + a))/c^2, -1/6*(3*(2*c*d^3 - 3*a*d*e^2)*sqrt(-c)*arctan(sqrt(-c)*
x/sqrt(c*x^2 + a)) - (2*c*e^3*x^2 + 9*c*d*e^2*x + 18*c*d^2*e - 4*a*e^3)*sqrt(c*x^2 + a))/c^2]

________________________________________________________________________________________

Sympy [A]  time = 5.99054, size = 216, normalized size = 1.96 \begin{align*} \frac{3 \sqrt{a} d e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{3 a d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{3} \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 ) + 3 d^{2} 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 ) + e^{3} \left (\begin{cases} - \frac{2 a \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{x^{2} \sqrt{a + c x^{2}}}{3 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+a)**(1/2),x)

[Out]

3*sqrt(a)*d*e**2*x*sqrt(1 + c*x**2/a)/(2*c) - 3*a*d*e**2*asinh(sqrt(c)*x/sqrt(a))/(2*c**(3/2)) + d**3*Piecewis
e((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))) + 3*d**2*e*Piecewise((x**2/(2*sqrt(a)
), Eq(c, 0)), (sqrt(a + c*x**2)/c, True)) + e**3*Piecewise((-2*a*sqrt(a + c*x**2)/(3*c**2) + x**2*sqrt(a + c*x
**2)/(3*c), Ne(c, 0)), (x**4/(4*sqrt(a)), True))

________________________________________________________________________________________

Giac [A]  time = 1.36014, size = 122, normalized size = 1.11 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2} + a}{\left (x{\left (\frac{2 \, x e^{3}}{c} + \frac{9 \, d e^{2}}{c}\right )} + \frac{2 \,{\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (2 \, c d^{3} - 3 \, a d 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)^3/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(c*x^2 + a)*(x*(2*x*e^3/c + 9*d*e^2/c) + 2*(9*c^2*d^2*e - 2*a*c*e^3)/c^3) - 1/2*(2*c*d^3 - 3*a*d*e^2)*
log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)