### 3.579 $$\int \frac{(d+e x)^3}{(a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=79 $-\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0267251, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {723, 637} $-\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(a + c*x^2)^(5/2),x]

[Out]

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

Rule 723

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[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{\left (2 \left (c d^2+a e^2\right )\right ) \int \frac{d+e x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac{2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.104651, size = 78, normalized size = 0.99 $\frac{-3 a^2 c e \left (d^2+e^2 x^2\right )-2 a^3 e^3+3 a c^2 d x \left (d^2+e^2 x^2\right )+2 c^3 d^3 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3/(a + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.045, size = 83, normalized size = 1.1 \begin{align*} -{\frac{-3\,a{c}^{2}d{e}^{2}{x}^{3}-2\,{c}^{3}{d}^{3}{x}^{3}+3\,{e}^{3}{x}^{2}{a}^{2}c-3\,{d}^{3}xa{c}^{2}+2\,{a}^{3}{e}^{3}+3\,{a}^{2}c{d}^{2}e}{3\,{a}^{2}{c}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.70726, size = 180, normalized size = 2.28 \begin{align*} -\frac{e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{2 \, d^{3} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{3} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{d e^{2} x}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{d e^{2} x}{\sqrt{c x^{2} + a} a c} - \frac{d^{2} e}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} - \frac{2 \, a e^{3}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.25374, size = 213, normalized size = 2.7 \begin{align*} -\frac{{\left (3 \, a^{2} c e^{3} x^{2} - 3 \, a c^{2} d^{3} x + 3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} -{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} 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)^(5/2),x, algorithm="fricas")

[Out]

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

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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{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**3/(a + c*x**2)**(5/2), x)

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Giac [A]  time = 1.27188, size = 119, normalized size = 1.51 \begin{align*} \frac{{\left (\frac{3 \, d^{3}}{a} - x{\left (\frac{3 \, e^{3}}{c} - \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )}\right )} x - \frac{3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{3 \,{\left (c x^{2} + a\right )}^{\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)^(5/2),x, algorithm="giac")

[Out]

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