### 3.597 $$\int \frac{a+c x^2}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=61 $-\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0225897, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $-\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/(d + e*x)^(7/2),x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{a+c x^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{7/2}}-\frac{2 c d}{e^2 (d+e x)^{5/2}}+\frac{c}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{5 e^3 (d+e x)^{5/2}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0385796, size = 44, normalized size = 0.72 $-\frac{2 \left (3 a e^2+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(3*a*e^2 + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 41, normalized size = 0.7 \begin{align*} -{\frac{30\,c{e}^{2}{x}^{2}+40\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(15*c*e^2*x^2+20*c*d*e*x+3*a*e^2+8*c*d^2)/e^3

________________________________________________________________________________________

Maxima [A]  time = 1.09602, size = 59, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c - 10 \,{\left (e x + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*c - 10*(e*x + d)*c*d + 3*c*d^2 + 3*a*e^2)/((e*x + d)^(5/2)*e^3)

________________________________________________________________________________________

Fricas [A]  time = 1.82718, size = 158, normalized size = 2.59 \begin{align*} -\frac{2 \,{\left (15 \, c e^{2} x^{2} + 20 \, c d e x + 8 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(15*c*e^2*x^2 + 20*c*d*e*x + 8*c*d^2 + 3*a*e^2)*sqrt(e*x + d)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3
*e^3)

________________________________________________________________________________________

Sympy [A]  time = 2.8407, size = 252, normalized size = 4.13 \begin{align*} \begin{cases} - \frac{6 a e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 1
6*c*d**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*c*d*e*x/(1
5*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*c*e**2*x**2/(15*d**2*
e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a*x + c*x**3/3)/d**
(7/2), True))

________________________________________________________________________________________

Giac [A]  time = 1.35545, size = 61, normalized size = 1. \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} c - 10 \,{\left (x e + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^2*c - 10*(x*e + d)*c*d + 3*c*d^2 + 3*a*e^2)*e^(-3)/(x*e + d)^(5/2)