3.5.88 \(\int \frac {1}{(a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}})^{5/2}} \, dx\) [488]

Optimal. Leaf size=410 \[ \frac {3 b^7 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^4}-\frac {7 b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^3}+\frac {63 b^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^2}-\frac {105 b^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )}+\frac {45 b^2 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {15 b \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^6 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {105 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \]

[Out]

3/4*b^7*(a+b/x^(1/3))/a^8/(b+a*x^(1/3))^4/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)-7*b^6*(a+b/x^(1/3))/a^8/(b+a*x
^(1/3))^3/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)+63/2*b^5*(a+b/x^(1/3))/a^8/(b+a*x^(1/3))^2/(a^2+b^2/x^(2/3)+2*
a*b/x^(1/3))^(1/2)-105*b^4*(a+b/x^(1/3))/a^8/(b+a*x^(1/3))/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)+45*b^2*(a+b/x
^(1/3))*x^(1/3)/a^7/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)-15/2*b*(a+b/x^(1/3))*x^(2/3)/a^6/(a^2+b^2/x^(2/3)+2*
a*b/x^(1/3))^(1/2)+(a+b/x^(1/3))*x/a^5/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)-105*b^3*(a+b/x^(1/3))*ln(b+a*x^(1
/3))/a^8/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \begin {gather*} \frac {3 b^7 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac {7 b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac {63 b^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac {105 b^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac {105 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {45 b^2 \sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^7 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {15 b x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {x \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-5/2),x]

[Out]

(3*b^7*(a + b/x^(1/3)))/(4*a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))^4) - (7*b^6*(a + b/x^
(1/3)))/(a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))^3) + (63*b^5*(a + b/x^(1/3)))/(2*a^8*Sq
rt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))^2) - (105*b^4*(a + b/x^(1/3)))/(a^8*Sqrt[a^2 + b^2/x^(
2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))) + (45*b^2*(a + b/x^(1/3))*x^(1/3))/(a^7*Sqrt[a^2 + b^2/x^(2/3) + (2*a
*b)/x^(1/3)]) - (15*b*(a + b/x^(1/3))*x^(2/3))/(2*a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) + ((a + b/x^(
1/3))*x)/(a^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) - (105*b^3*(a + b/x^(1/3))*Log[b + a*x^(1/3)])/(a^8*S
qrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {1}{\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2}} \, dx &=3 \text {Subst}\left (\int \frac {x^2}{\left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b+\frac {b^2}{x}\right )^5} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \frac {x^7}{\left (b^2+a b x\right )^5} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \left (\frac {15}{a^7 b^3}-\frac {5 x}{a^6 b^4}+\frac {x^2}{a^5 b^5}-\frac {b^2}{a^7 (b+a x)^5}+\frac {7 b}{a^7 (b+a x)^4}-\frac {21}{a^7 (b+a x)^3}+\frac {35}{a^7 b (b+a x)^2}-\frac {35}{a^7 b^2 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {3 \left (a b^7+\frac {b^8}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^4}-\frac {7 \left (a b^6+\frac {b^7}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^3}+\frac {63 \left (a b^5+\frac {b^6}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^2}-\frac {105 \left (a b^4+\frac {b^5}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )}+\frac {45 \left (a b^2+\frac {b^3}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {15 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^6 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {105 \left (a b^3+\frac {b^4}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 152, normalized size = 0.37 \begin {gather*} \frac {\left (b+a \sqrt [3]{x}\right ) \left (-319 b^7-856 a b^6 \sqrt [3]{x}-444 a^2 b^5 x^{2/3}+544 a^3 b^4 x+556 a^4 b^3 x^{4/3}+84 a^5 b^2 x^{5/3}-14 a^6 b x^2+4 a^7 x^{7/3}-420 b^3 \left (b+a \sqrt [3]{x}\right )^4 \log \left (b+a \sqrt [3]{x}\right )\right )}{4 a^8 \left (\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}\right )^{5/2} x^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-5/2),x]

[Out]

((b + a*x^(1/3))*(-319*b^7 - 856*a*b^6*x^(1/3) - 444*a^2*b^5*x^(2/3) + 544*a^3*b^4*x + 556*a^4*b^3*x^(4/3) + 8
4*a^5*b^2*x^(5/3) - 14*a^6*b*x^2 + 4*a^7*x^(7/3) - 420*b^3*(b + a*x^(1/3))^4*Log[b + a*x^(1/3)]))/(4*a^8*((b +
a*x^(1/3))^2/x^(2/3))^(5/2)*x^(5/3))

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Maple [A]
time = 0.04, size = 199, normalized size = 0.49

method	result	size
derivativedivides	\(-\frac {\left (-4 a^{7} x^{\frac {7}{3}}+14 a^{6} b \,x^{2}+420 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{4} b^{3} x^{\frac {4}{3}}-84 a^{5} b^{2} x^{\frac {5}{3}}+1680 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{3} b^{4} x -556 x^{\frac {4}{3}} b^{3} a^{4}+2520 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{2} b^{5} x^{\frac {2}{3}}-544 a^{3} b^{4} x +1680 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a \,b^{6} x^{\frac {1}{3}}+444 a^{2} b^{5} x^{\frac {2}{3}}+420 \ln \left (b +a \,x^{\frac {1}{3}}\right ) b^{7}+856 a \,b^{6} x^{\frac {1}{3}}+319 b^{7}\right ) \left (b +a \,x^{\frac {1}{3}}\right )}{4 a^{8} x^{\frac {5}{3}} \left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}}}\)	\(199\)
default	\(\frac {\left (4 a^{7} x^{\frac {7}{3}}+84 a^{5} b^{2} x^{\frac {5}{3}}-420 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{4} b^{3} x^{\frac {4}{3}}+556 x^{\frac {4}{3}} b^{3} a^{4}-2520 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{2} b^{5} x^{\frac {2}{3}}-444 a^{2} b^{5} x^{\frac {2}{3}}-1680 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a \,b^{6} x^{\frac {1}{3}}-1680 \ln \left (b +a \,x^{\frac {1}{3}}\right ) a^{3} b^{4} x -14 a^{6} b \,x^{2}-856 a \,b^{6} x^{\frac {1}{3}}-420 \ln \left (b +a \,x^{\frac {1}{3}}\right ) b^{7}+544 a^{3} b^{4} x -319 b^{7}\right ) \left (b +a \,x^{\frac {1}{3}}\right )}{4 \left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}} x^{\frac {5}{3}} a^{8}}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/4/((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(5/2)/x^(5/3)*(4*a^7*x^(7/3)+84*a^5*b^2*x^(5/3)-420*ln(b+a*x^(1/
3))*a^4*b^3*x^(4/3)+556*x^(4/3)*b^3*a^4-2520*ln(b+a*x^(1/3))*a^2*b^5*x^(2/3)-444*a^2*b^5*x^(2/3)-1680*ln(b+a*x
^(1/3))*a*b^6*x^(1/3)-1680*ln(b+a*x^(1/3))*a^3*b^4*x-14*a^6*b*x^2-856*a*b^6*x^(1/3)-420*ln(b+a*x^(1/3))*b^7+54
4*a^3*b^4*x-319*b^7)*(b+a*x^(1/3))/a^8

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Maxima [A]
time = 0.29, size = 139, normalized size = 0.34 \begin {gather*} \frac {4 \, a^{7} x^{\frac {7}{3}} - 14 \, a^{6} b x^{2} + 84 \, a^{5} b^{2} x^{\frac {5}{3}} + 556 \, a^{4} b^{3} x^{\frac {4}{3}} + 544 \, a^{3} b^{4} x - 444 \, a^{2} b^{5} x^{\frac {2}{3}} - 856 \, a b^{6} x^{\frac {1}{3}} - 319 \, b^{7}}{4 \, {\left (a^{12} x^{\frac {4}{3}} + 4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac {2}{3}} + 4 \, a^{9} b^{3} x^{\frac {1}{3}} + a^{8} b^{4}\right )}} - \frac {105 \, b^{3} \log \left (a x^{\frac {1}{3}} + b\right )}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x, algorithm="maxima")

[Out]

1/4*(4*a^7*x^(7/3) - 14*a^6*b*x^2 + 84*a^5*b^2*x^(5/3) + 556*a^4*b^3*x^(4/3) + 544*a^3*b^4*x - 444*a^2*b^5*x^(
2/3) - 856*a*b^6*x^(1/3) - 319*b^7)/(a^12*x^(4/3) + 4*a^11*b*x + 6*a^10*b^2*x^(2/3) + 4*a^9*b^3*x^(1/3) + a^8*
b^4) - 105*b^3*log(a*x^(1/3) + b)/a^8

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(5/2),x)

[Out]

Integral((a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(-5/2), x)

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Giac [A]
time = 4.70, size = 147, normalized size = 0.36 \begin {gather*} -\frac {105 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{8} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} - \frac {420 \, a^{3} b^{4} x + 1134 \, a^{2} b^{5} x^{\frac {2}{3}} + 1036 \, a b^{6} x^{\frac {1}{3}} + 319 \, b^{7}}{4 \, {\left (a x^{\frac {1}{3}} + b\right )}^{4} a^{8} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} + \frac {2 \, a^{10} x - 15 \, a^{9} b x^{\frac {2}{3}} + 90 \, a^{8} b^{2} x^{\frac {1}{3}}}{2 \, a^{15} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x, algorithm="giac")

[Out]

-105*b^3*log(abs(a*x^(1/3) + b))/(a^8*sgn(a*x + b*x^(2/3))*sgn(x)) - 1/4*(420*a^3*b^4*x + 1134*a^2*b^5*x^(2/3)
+ 1036*a*b^6*x^(1/3) + 319*b^7)/((a*x^(1/3) + b)^4*a^8*sgn(a*x + b*x^(2/3))*sgn(x)) + 1/2*(2*a^10*x - 15*a^9*
b*x^(2/3) + 90*a^8*b^2*x^(1/3))/(a^15*sgn(a*x + b*x^(2/3))*sgn(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(5/2),x)

[Out]

int(1/(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(5/2), x)

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