3.795 \(\int \frac{(a+c x^4)^{3/2}}{x^{19}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{4 c^2 \left (a+c x^4\right )^{5/2}}{315 a^3 x^{10}}+\frac{2 c \left (a+c x^4\right )^{5/2}}{63 a^2 x^{14}}-\frac{\left (a+c x^4\right )^{5/2}}{18 a x^{18}} \]

[Out]

-(a + c*x^4)^(5/2)/(18*a*x^18) + (2*c*(a + c*x^4)^(5/2))/(63*a^2*x^14) - (4*c^2*(a + c*x^4)^(5/2))/(315*a^3*x^
10)

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Rubi [A]  time = 0.0188366, antiderivative size = 68, 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, Rules used = {271, 264} \[ -\frac{4 c^2 \left (a+c x^4\right )^{5/2}}{315 a^3 x^{10}}+\frac{2 c \left (a+c x^4\right )^{5/2}}{63 a^2 x^{14}}-\frac{\left (a+c x^4\right )^{5/2}}{18 a x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^19,x]

[Out]

-(a + c*x^4)^(5/2)/(18*a*x^18) + (2*c*(a + c*x^4)^(5/2))/(63*a^2*x^14) - (4*c^2*(a + c*x^4)^(5/2))/(315*a^3*x^
10)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx &=-\frac{\left (a+c x^4\right )^{5/2}}{18 a x^{18}}-\frac{(4 c) \int \frac{\left (a+c x^4\right )^{3/2}}{x^{15}} \, dx}{9 a}\\ &=-\frac{\left (a+c x^4\right )^{5/2}}{18 a x^{18}}+\frac{2 c \left (a+c x^4\right )^{5/2}}{63 a^2 x^{14}}+\frac{\left (8 c^2\right ) \int \frac{\left (a+c x^4\right )^{3/2}}{x^{11}} \, dx}{63 a^2}\\ &=-\frac{\left (a+c x^4\right )^{5/2}}{18 a x^{18}}+\frac{2 c \left (a+c x^4\right )^{5/2}}{63 a^2 x^{14}}-\frac{4 c^2 \left (a+c x^4\right )^{5/2}}{315 a^3 x^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0119153, size = 42, normalized size = 0.62 \[ -\frac{\left (a+c x^4\right )^{5/2} \left (35 a^2-20 a c x^4+8 c^2 x^8\right )}{630 a^3 x^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^19,x]

[Out]

-((a + c*x^4)^(5/2)*(35*a^2 - 20*a*c*x^4 + 8*c^2*x^8))/(630*a^3*x^18)

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{c}^{2}{x}^{8}-20\,c{x}^{4}a+35\,{a}^{2}}{630\,{x}^{18}{a}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^19,x)

[Out]

-1/630*(c*x^4+a)^(5/2)*(8*c^2*x^8-20*a*c*x^4+35*a^2)/x^18/a^3

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Maxima [A]  time = 0.962556, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{63 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} c^{2}}{x^{10}} - \frac{90 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} c}{x^{14}} + \frac{35 \,{\left (c x^{4} + a\right )}^{\frac{9}{2}}}{x^{18}}}{630 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="maxima")

[Out]

-1/630*(63*(c*x^4 + a)^(5/2)*c^2/x^10 - 90*(c*x^4 + a)^(7/2)*c/x^14 + 35*(c*x^4 + a)^(9/2)/x^18)/a^3

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Fricas [A]  time = 1.68169, size = 139, normalized size = 2.04 \begin{align*} -\frac{{\left (8 \, c^{4} x^{16} - 4 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} + 50 \, a^{3} c x^{4} + 35 \, a^{4}\right )} \sqrt{c x^{4} + a}}{630 \, a^{3} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="fricas")

[Out]

-1/630*(8*c^4*x^16 - 4*a*c^3*x^12 + 3*a^2*c^2*x^8 + 50*a^3*c*x^4 + 35*a^4)*sqrt(c*x^4 + a)/(a^3*x^18)

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Sympy [B]  time = 7.50368, size = 420, normalized size = 6.18 \begin{align*} - \frac{35 a^{6} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{120 a^{5} c^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{138 a^{4} c^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{52 a^{3} c^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{3 a^{2} c^{\frac{17}{2}} x^{16} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{12 a c^{\frac{19}{2}} x^{20} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac{8 c^{\frac{21}{2}} x^{24} \sqrt{\frac{a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**19,x)

[Out]

-35*a**6*c**(9/2)*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x**24) - 12
0*a**5*c**(11/2)*x**4*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x**24)
- 138*a**4*c**(13/2)*x**8*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x**
24) - 52*a**3*c**(15/2)*x**12*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6
*x**24) - 3*a**2*c**(17/2)*x**16*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c
**6*x**24) - 12*a*c**(19/2)*x**20*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*
c**6*x**24) - 8*c**(21/2)*x**24*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c*
*6*x**24)

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Giac [A]  time = 1.09694, size = 143, normalized size = 2.1 \begin{align*} -\frac{\frac{3 \,{\left (15 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{7}{2}} - 42 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{2}\right )} c}{a^{2}} + \frac{35 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{9}{2}} - 135 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{7}{2}} c + 189 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{3}}{a^{2}}}{630 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="giac")

[Out]

-1/630*(3*(15*(c + a/x^4)^(7/2) - 42*(c + a/x^4)^(5/2)*c + 35*(c + a/x^4)^(3/2)*c^2)*c/a^2 + (35*(c + a/x^4)^(
9/2) - 135*(c + a/x^4)^(7/2)*c + 189*(c + a/x^4)^(5/2)*c^2 - 105*(c + a/x^4)^(3/2)*c^3)/a^2)/a