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

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

[Out]

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

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Rubi [A]  time = 0.046406, 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} \[ -\frac{4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt{d+e x}}+\frac{8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0636942, size = 96, normalized size = 0.78 \[ -\frac{2 \left (3 a^2 e^4+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+c^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a^2*e^4 + 2*a*c*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40
*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))

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Maple [A]  time = 0.044, size = 106, normalized size = 0.9 \begin{align*} -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}+80\,{c}^{2}d{x}^{3}{e}^{3}+60\,ac{e}^{4}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+80\,acd{e}^{3}x+640\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+32\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \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)^2/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*c^2*e^4*x^4+40*c^2*d*e^3*x^3+30*a*c*e^4*x^2+240*c^2*d^2*e^2*x^2+40*a*c*d*e^3*x+320*c^2
*d^3*e*x+3*a^2*e^4+16*a*c*d^2*e^2+128*c^2*d^4)/e^5

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Maxima [A]  time = 1.1731, size = 163, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 12 \, \sqrt{e x + d} c^{2} d\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 30 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{2} - 20 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{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)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*c^2 - 12*sqrt(e*x + d)*c^2*d)/e^4 - (3*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4 + 30*(3*c^
2*d^2 + a*c*e^2)*(e*x + d)^2 - 20*(c^2*d^3 + a*c*d*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^4))/e

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Fricas [A]  time = 1.86978, size = 289, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} - 128 \, c^{2} d^{4} - 16 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 30 \,{\left (8 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} - 40 \,{\left (8 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c^2*e^4*x^4 - 40*c^2*d*e^3*x^3 - 128*c^2*d^4 - 16*a*c*d^2*e^2 - 3*a^2*e^4 - 30*(8*c^2*d^2*e^2 + a*c*e^
4)*x^2 - 40*(8*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 3.14516, size = 592, normalized size = 4.81 \begin{align*} \begin{cases} - \frac{6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{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)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 32*a*c*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*
a*c*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 60*a*c*e*
*4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*c**2*d**2*e**2*x**2/(1
5*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(15*d**2
*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + 2*a*c*x**
3/3 + c**2*x**5/5)/d**(7/2), True))

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Giac [A]  time = 2.10165, size = 176, normalized size = 1.43 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\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)^2/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10)*e^(-15) - 2/15*(90*(x*e + d)^2*c^2*d^2 - 20*(x*e
+ d)*c^2*d^3 + 3*c^2*d^4 + 30*(x*e + d)^2*a*c*e^2 - 20*(x*e + d)*a*c*d*e^2 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*e^(-5)
/(x*e + d)^(5/2)