### 3.603 $$\int \frac{(a+c x^2)^2}{(d+e x)^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0472387, 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} $\frac{4 c \sqrt{d+e x} \left (a e^2+3 c d^2\right )}{e^5}+\frac{8 c d \left (a e^2+c d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}-\frac{8 c^2 d (d+e x)^{3/2}}{3 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0571416, size = 96, normalized size = 0.78 $\frac{2 \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d
*e^3*x^3 + 3*e^4*x^4)))/(15*e^5*(d + e*x)^(3/2))

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Maple [A]  time = 0.044, size = 106, normalized size = 0.9 \begin{align*} -{\frac{-6\,{c}^{2}{x}^{4}{e}^{4}+16\,{c}^{2}d{x}^{3}{e}^{3}-60\,ac{e}^{4}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-240\,acd{e}^{3}x-384\,{c}^{2}{d}^{3}ex+10\,{a}^{2}{e}^{4}-160\,ac{d}^{2}{e}^{2}-256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/(e*x+d)^(3/2)*(-3*c^2*e^4*x^4+8*c^2*d*e^3*x^3-30*a*c*e^4*x^2-48*c^2*d^2*e^2*x^2-120*a*c*d*e^3*x-192*c^2*
d^3*e*x+5*a^2*e^4-80*a*c*d^2*e^2-128*c^2*d^4)/e^5

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Maxima [A]  time = 1.18341, size = 161, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} - 20 \,{\left (e x + d\right )}^{\frac{3}{2}} c^{2} d + 30 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 12 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(e*x + d)^(5/2)*c^2 - 20*(e*x + d)^(3/2)*c^2*d + 30*(3*c^2*d^2 + a*c*e^2)*sqrt(e*x + d))/e^4 - 5*(c^2
*d^4 + 2*a*c*d^2*e^2 + a^2*e^4 - 12*(c^2*d^3 + a*c*d*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^4))/e

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Fricas [A]  time = 1.77229, size = 270, normalized size = 2.2 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{4} x^{4} - 8 \, c^{2} d e^{3} x^{3} + 128 \, c^{2} d^{4} + 80 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4} + 6 \,{\left (8 \, c^{2} d^{2} e^{2} + 5 \, a c e^{4}\right )} x^{2} + 24 \,{\left (8 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*c^2*e^4*x^4 - 8*c^2*d*e^3*x^3 + 128*c^2*d^4 + 80*a*c*d^2*e^2 - 5*a^2*e^4 + 6*(8*c^2*d^2*e^2 + 5*a*c*e^
4)*x^2 + 24*(8*c^2*d^3*e + 5*a*c*d*e^3)*x)*sqrt(e*x + d)/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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Sympy [A]  time = 19.936, size = 121, normalized size = 0.98 \begin{align*} - \frac{8 c^{2} d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{8 c d \left (a e^{2} + c d^{2}\right )}{e^{5} \sqrt{d + e x}} + \frac{\sqrt{d + e x} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{e^{5}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32235, size = 180, normalized size = 1.46 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e^{20} + 90 \, \sqrt{x e + d} c^{2} d^{2} e^{20} + 30 \, \sqrt{x e + d} a c e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} + 12 \,{\left (x e + d\right )} a c d e^{2} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*e^20 - 20*(x*e + d)^(3/2)*c^2*d*e^20 + 90*sqrt(x*e + d)*c^2*d^2*e^20 + 30*sqrt(x*e
+ d)*a*c*e^22)*e^(-25) + 2/3*(12*(x*e + d)*c^2*d^3 - c^2*d^4 + 12*(x*e + d)*a*c*d*e^2 - 2*a*c*d^2*e^2 - a^2*e
^4)*e^(-5)/(x*e + d)^(3/2)