### 3.570 $$\int \frac{(d+e x)^3}{(a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=106 $-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0574783, antiderivative size = 106, 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, 780, 217, 206} $-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(a + c*x^2)^(3/2),x]

[Out]

-(((a*e - c*d*x)*(d + e*x)^2)/(a*c*Sqrt[a + c*x^2])) - (e*(2*(c*d^2 - a*e^2) + c*d*e*x)*Sqrt[a + c*x^2])/(a*c^
2) + (3*d*e^2*ArcTanh[(Sqrt[c]*x)/Sqrt[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 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}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}+\frac{\int \frac{(d+e x) \left (2 a e^2-2 c d e x\right )}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{\left (3 d e^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{\left (3 d e^2\right ) \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)^2}{a c \sqrt{a+c x^2}}-\frac{e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt{a+c x^2}}{a c^2}+\frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.10542, size = 91, normalized size = 0.86 $\frac{2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x}{a c^2 \sqrt{a+c x^2}}+\frac{3 d e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 118, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+a}}}+3\,{\frac{d{e}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+a}}}+{\frac{{d}^{3}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)^3/(c*x^2+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.96924, size = 531, normalized size = 5.01 \begin{align*} \left [\frac{3 \,{\left (a c d e^{2} x^{2} + a^{2} 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 (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{3 \,{\left (a c d e^{2} x^{2} + a^{2} d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d 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)^3/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(3*(a*c*d*e^2*x^2 + a^2*d*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(a*c*e^3*x^2 -
3*a*c*d^2*e + 2*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(a*c^3*x^2 + a^2*c^2), -(3*(a*c*d*e^2*x
^2 + a^2*d*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (a*c*e^3*x^2 - 3*a*c*d^2*e + 2*a^2*e^3 + (c^2*d^
3 - 3*a*c*d*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 )^{3}}{\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)**3/(c*x**2+a)**(3/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.55404, size = 135, normalized size = 1.27 \begin{align*} -\frac{3 \, d e^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} + \frac{x{\left (\frac{x e^{3}}{c} + \frac{c^{3} d^{3} - 3 \, a c^{2} d e^{2}}{a c^{3}}\right )} - \frac{3 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}}{a c^{3}}}{\sqrt{c x^{2} + a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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