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]

-1/18*(c*x^4+a)^(5/2)/a/x^18+2/63*c*(c*x^4+a)^(5/2)/a^2/x^14-4/315*c^2*(c*x^4+a)^(5/2)/a^3/x^10

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Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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]

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

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fricas [A]  time = 0.56, size = 60, normalized size = 0.88 \[ -\frac {{\left (8 \, c^{4} x^{16} - 4 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} + 50 \, a^{3} c x^{4} + 35 \, a^{4}\right )} \sqrt {c x^{4} + a}}{630 \, a^{3} x^{18}} \]

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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giac [B]  time = 0.25, size = 206, normalized size = 3.03 \[ \frac {8 \, {\left (210 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{12} c^{\frac {9}{2}} + 315 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{10} a c^{\frac {9}{2}} + 441 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} a^{2} c^{\frac {9}{2}} + 126 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} a^{3} c^{\frac {9}{2}} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{4} c^{\frac {9}{2}} - 9 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{5} c^{\frac {9}{2}} + a^{6} c^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/315*(210*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^12*c^(9/2) + 315*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^10*a*c^(9/2) + 441
*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^8*a^2*c^(9/2) + 126*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^6*a^3*c^(9/2) + 36*(sqrt(
c)*x^2 - sqrt(c*x^4 + a))^4*a^4*c^(9/2) - 9*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^2*a^5*c^(9/2) + a^6*c^(9/2))/((sqr
t(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^9

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maple [A]  time = 0.01, size = 39, normalized size = 0.57 \[ -\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (8 c^{2} x^{8}-20 a c \,x^{4}+35 a^{2}\right )}{630 a^{3} x^{18}} \]

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 = 1.33, size = 52, normalized size = 0.76 \[ -\frac {\frac {63 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} c^{2}}{x^{10}} - \frac {90 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} c}{x^{14}} + \frac {35 \, {\left (c x^{4} + a\right )}^{\frac {9}{2}}}{x^{18}}}{630 \, a^{3}} \]

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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mupad [B]  time = 2.07, size = 91, normalized size = 1.34 \[ \frac {2\,c^3\,\sqrt {c\,x^4+a}}{315\,a^2\,x^6}-\frac {5\,c\,\sqrt {c\,x^4+a}}{63\,x^{14}}-\frac {4\,c^4\,\sqrt {c\,x^4+a}}{315\,a^3\,x^2}-\frac {a\,\sqrt {c\,x^4+a}}{18\,x^{18}}-\frac {c^2\,\sqrt {c\,x^4+a}}{210\,a\,x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(3/2)/x^19,x)

[Out]

(2*c^3*(a + c*x^4)^(1/2))/(315*a^2*x^6) - (5*c*(a + c*x^4)^(1/2))/(63*x^14) - (4*c^4*(a + c*x^4)^(1/2))/(315*a
^3*x^2) - (a*(a + c*x^4)^(1/2))/(18*x^18) - (c^2*(a + c*x^4)^(1/2))/(210*a*x^10)

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sympy [B]  time = 5.41, size = 420, normalized size = 6.18 \[ - \frac {35 a^{6} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {120 a^{5} c^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {138 a^{4} c^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {52 a^{3} c^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {3 a^{2} c^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {12 a c^{\frac {19}{2}} x^{20} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {8 c^{\frac {21}{2}} x^{24} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} \]

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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