### 3.602 $$\int \frac{(a+c x^2)^2}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=123 $\frac{4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{8 c d \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac{2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5}$

[Out]

(-2*(c*d^2 + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (8*c*d*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^5 + (4*c*(3*c*d^2 + a*e^2
)*(d + e*x)^(3/2))/(3*e^5) - (8*c^2*d*(d + e*x)^(5/2))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0472858, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {697} $\frac{4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{8 c d \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac{2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(-2*(c*d^2 + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (8*c*d*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^5 + (4*c*(3*c*d^2 + a*e^2
)*(d + e*x)^(3/2))/(3*e^5) - (8*c^2*d*(d + e*x)^(5/2))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)

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{\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{3/2}}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 c \left (3 c d^2+a e^2\right ) \sqrt{d+e x}}{e^4}-\frac{4 c^2 d (d+e x)^{3/2}}{e^4}+\frac{c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{8 c d \left (c d^2+a e^2\right ) \sqrt{d+e x}}{e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0559045, size = 97, normalized size = 0.79 $-\frac{2 \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )}{105 e^5 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(105*a^2*e^4 + 70*a*c*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + 3*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d
*e^3*x^3 - 5*e^4*x^4)))/(105*e^5*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 106, normalized size = 0.9 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}+48\,{c}^{2}d{x}^{3}{e}^{3}-140\,ac{e}^{4}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+560\,acd{e}^{3}x+384\,{c}^{2}{d}^{3}ex+210\,{a}^{2}{e}^{4}+1120\,ac{d}^{2}{e}^{2}+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105/(e*x+d)^(1/2)*(-15*c^2*e^4*x^4+24*c^2*d*e^3*x^3-70*a*c*e^4*x^2-48*c^2*d^2*e^2*x^2+280*a*c*d*e^3*x+192*c
^2*d^3*e*x+105*a^2*e^4+560*a*c*d^2*e^2+384*c^2*d^4)/e^5

________________________________________________________________________________________

Maxima [A]  time = 1.10377, size = 163, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 84 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} d + 70 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 420 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((15*(e*x + d)^(7/2)*c^2 - 84*(e*x + d)^(5/2)*c^2*d + 70*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(3/2) - 420*(c^
2*d^3 + a*c*d*e^2)*sqrt(e*x + d))/e^4 - 105*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^4))/e

________________________________________________________________________________________

Fricas [A]  time = 1.69958, size = 261, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 24 \, c^{2} d e^{3} x^{3} - 384 \, c^{2} d^{4} - 560 \, a c d^{2} e^{2} - 105 \, a^{2} e^{4} + 2 \,{\left (24 \, c^{2} d^{2} e^{2} + 35 \, a c e^{4}\right )} x^{2} - 8 \,{\left (24 \, c^{2} d^{3} e + 35 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*c^2*e^4*x^4 - 24*c^2*d*e^3*x^3 - 384*c^2*d^4 - 560*a*c*d^2*e^2 - 105*a^2*e^4 + 2*(24*c^2*d^2*e^2 + 3
5*a*c*e^4)*x^2 - 8*(24*c^2*d^3*e + 35*a*c*d*e^3)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

________________________________________________________________________________________

Sympy [A]  time = 12.2785, size = 126, normalized size = 1.02 \begin{align*} - \frac{8 c^{2} d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (- 8 a c d e^{2} - 8 c^{2} d^{3}\right )}{e^{5}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2}}{e^{5} \sqrt{d + e x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*c**2*d*(d + e*x)**(5/2)/(5*e**5) + 2*c**2*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(3/2)*(4*a*c*e**2 + 12*c**
2*d**2)/(3*e**5) + sqrt(d + e*x)*(-8*a*c*d*e**2 - 8*c**2*d**3)/e**5 - 2*(a*e**2 + c*d**2)**2/(e**5*sqrt(d + e*
x))

________________________________________________________________________________________

Giac [A]  time = 1.30337, size = 185, normalized size = 1.5 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c e^{32} - 420 \, \sqrt{x e + d} a c d e^{32}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*e^30 - 84*(x*e + d)^(5/2)*c^2*d*e^30 + 210*(x*e + d)^(3/2)*c^2*d^2*e^30 - 420*sq
rt(x*e + d)*c^2*d^3*e^30 + 70*(x*e + d)^(3/2)*a*c*e^32 - 420*sqrt(x*e + d)*a*c*d*e^32)*e^(-35) - 2*(c^2*d^4 +
2*a*c*d^2*e^2 + a^2*e^4)*e^(-5)/sqrt(x*e + d)