3.782 \(\int x^7 \left (a+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac{a \left (a+c x^4\right )^{5/2}}{10 c^2} \]

[Out]

-(a*(a + c*x^4)^(5/2))/(10*c^2) + (a + c*x^4)^(7/2)/(14*c^2)

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Rubi [A]  time = 0.0599741, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac{a \left (a+c x^4\right )^{5/2}}{10 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + c*x^4)^(3/2),x]

[Out]

-(a*(a + c*x^4)^(5/2))/(10*c^2) + (a + c*x^4)^(7/2)/(14*c^2)

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Rubi in Sympy [A]  time = 7.1595, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + c x^{4}\right )^{\frac{5}{2}}}{10 c^{2}} + \frac{\left (a + c x^{4}\right )^{\frac{7}{2}}}{14 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(c*x**4+a)**(3/2),x)

[Out]

-a*(a + c*x**4)**(5/2)/(10*c**2) + (a + c*x**4)**(7/2)/(14*c**2)

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Mathematica [A]  time = 0.0338798, size = 28, normalized size = 0.74 \[ \frac{\left (a+c x^4\right )^{5/2} \left (5 c x^4-2 a\right )}{70 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + c*x^4)^(3/2),x]

[Out]

((a + c*x^4)^(5/2)*(-2*a + 5*c*x^4))/(70*c^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-5\,c{x}^{4}+2\,a}{70\,{c}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(c*x^4+a)^(3/2),x)

[Out]

-1/70*(c*x^4+a)^(5/2)*(-5*c*x^4+2*a)/c^2

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Maxima [A]  time = 1.4254, size = 41, normalized size = 1.08 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{14 \, c^{2}} - \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}} a}{10 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^7,x, algorithm="maxima")

[Out]

1/14*(c*x^4 + a)^(7/2)/c^2 - 1/10*(c*x^4 + a)^(5/2)*a/c^2

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Fricas [A]  time = 0.254951, size = 61, normalized size = 1.61 \[ \frac{{\left (5 \, c^{3} x^{12} + 8 \, a c^{2} x^{8} + a^{2} c x^{4} - 2 \, a^{3}\right )} \sqrt{c x^{4} + a}}{70 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^7,x, algorithm="fricas")

[Out]

1/70*(5*c^3*x^12 + 8*a*c^2*x^8 + a^2*c*x^4 - 2*a^3)*sqrt(c*x^4 + a)/c^2

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Sympy [A]  time = 9.4906, size = 83, normalized size = 2.18 \[ \begin{cases} - \frac{a^{3} \sqrt{a + c x^{4}}}{35 c^{2}} + \frac{a^{2} x^{4} \sqrt{a + c x^{4}}}{70 c} + \frac{4 a x^{8} \sqrt{a + c x^{4}}}{35} + \frac{c x^{12} \sqrt{a + c x^{4}}}{14} & \text{for}\: c \neq 0 \\\frac{a^{\frac{3}{2}} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(c*x**4+a)**(3/2),x)

[Out]

Piecewise((-a**3*sqrt(a + c*x**4)/(35*c**2) + a**2*x**4*sqrt(a + c*x**4)/(70*c)
+ 4*a*x**8*sqrt(a + c*x**4)/35 + c*x**12*sqrt(a + c*x**4)/14, Ne(c, 0)), (a**(3/
2)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.214462, size = 105, normalized size = 2.76 \[ \frac{\frac{7 \,{\left (3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a\right )} a}{c} + \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{c}}{210 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^7,x, algorithm="giac")

[Out]

1/210*(7*(3*(c*x^4 + a)^(5/2) - 5*(c*x^4 + a)^(3/2)*a)*a/c + (15*(c*x^4 + a)^(7/
2) - 42*(c*x^4 + a)^(5/2)*a + 35*(c*x^4 + a)^(3/2)*a^2)/c)/c