3.5.92 \(\int (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}})^{7/2} \, dx\) [492]

Optimal. Leaf size=391 \[ -\frac {6 b^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}}{\left (a+\frac {b}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac {126 a^2 b^5 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {105 a^3 b^4 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {70 a^4 b^3 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt {x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {63 a^5 b^2 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {42 a^6 b \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {a^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x}{a+\frac {b}{\sqrt [6]{x}}}+\frac {42 a b^6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \log \left (\sqrt [6]{x}\right )}{a+\frac {b}{\sqrt [6]{x}}} \]

[Out]

-6*b^7*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))/x^(1/6)+126*a^2*b^5*x^(1/6)*(a^2+b^2/x^(1/3)+2*a*b/
x^(1/6))^(1/2)/(a+b/x^(1/6))+105*a^3*b^4*x^(1/3)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+63/2*a^5*
b^2*x^(2/3)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+42/5*a^6*b*x^(5/6)*(a^2+b^2/x^(1/3)+2*a*b/x^(1
/6))^(1/2)/(a+b/x^(1/6))+a^7*x*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+7*a*b^6*ln(x)*(a^2+b^2/x^(1
/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+70*a^4*b^3*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)*x^(1/2)/(a+b/x^(1/6))

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Rubi [A]
time = 0.12, antiderivative size = 391, 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 {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {126 a^2 b^5 \sqrt [6]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {a^7 x \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {42 a^6 b x^{5/6} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {63 a^5 b^2 x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {70 a^4 b^3 \sqrt {x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {105 a^3 b^4 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(-6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)])/((a + b/x^(1/6))*x^(1/6)) + (126*a^2*b^5*Sqrt[a^2 + b^2/x^(
1/3) + (2*a*b)/x^(1/6)]*x^(1/6))/(a + b/x^(1/6)) + (105*a^3*b^4*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1
/3))/(a + b/x^(1/6)) + (70*a^4*b^3*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Sqrt[x])/(a + b/x^(1/6)) + (63*a^
5*b^2*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(2/3))/(2*(a + b/x^(1/6))) + (42*a^6*b*Sqrt[a^2 + b^2/x^(1/3
) + (2*a*b)/x^(1/6)]*x^(5/6))/(5*(a + b/x^(1/6))) + (a^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x)/(a + b/x
^(1/6)) + (42*a*b^6*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Log[x^(1/6)])/(a + b/x^(1/6))

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 \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx &=6 \text {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{7/2} x^5 \, dx,x,\sqrt [6]{x}\right )\\ &=\frac {\left (6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}\right ) \text {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^7 x^5 \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac {b^2}{\sqrt [6]{x}}\right )}\\ &=\frac {\left (6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}\right ) \text {Subst}\left (\int \frac {\left (b^2+a b x\right )^7}{x^2} \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac {b^2}{\sqrt [6]{x}}\right )}\\ &=\frac {\left (6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}\right ) \text {Subst}\left (\int \left (21 a^2 b^{12}+\frac {b^{14}}{x^2}+\frac {7 a b^{13}}{x}+35 a^3 b^{11} x+35 a^4 b^{10} x^2+21 a^5 b^9 x^3+7 a^6 b^8 x^4+a^7 b^7 x^5\right ) \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac {b^2}{\sqrt [6]{x}}\right )}\\ &=-\frac {6 b^8 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}}{\left (a b+\frac {b^2}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac {126 a^2 b^6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a b+\frac {b^2}{\sqrt [6]{x}}}+\frac {105 a^3 b^5 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a b+\frac {b^2}{\sqrt [6]{x}}}+\frac {70 a^4 b^4 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt {x}}{a b+\frac {b^2}{\sqrt [6]{x}}}+\frac {63 a^5 b^3 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a b+\frac {b^2}{\sqrt [6]{x}}\right )}+\frac {42 a^6 b^2 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a b+\frac {b^2}{\sqrt [6]{x}}\right )}+\frac {a^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x}{a+\frac {b}{\sqrt [6]{x}}}+\frac {7 a b^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \log (x)}{a b+\frac {b^2}{\sqrt [6]{x}}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 124, normalized size = 0.32 \begin {gather*} \frac {\sqrt {\frac {\left (b+a \sqrt [6]{x}\right )^2}{\sqrt [3]{x}}} \left (-60 b^7+1260 a^2 b^5 \sqrt [3]{x}+1050 a^3 b^4 \sqrt {x}+700 a^4 b^3 x^{2/3}+315 a^5 b^2 x^{5/6}+84 a^6 b x+10 a^7 x^{7/6}+70 a b^6 \sqrt [6]{x} \log (x)\right )}{10 \left (b+a \sqrt [6]{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(Sqrt[(b + a*x^(1/6))^2/x^(1/3)]*(-60*b^7 + 1260*a^2*b^5*x^(1/3) + 1050*a^3*b^4*Sqrt[x] + 700*a^4*b^3*x^(2/3)
+ 315*a^5*b^2*x^(5/6) + 84*a^6*b*x + 10*a^7*x^(7/6) + 70*a*b^6*x^(1/6)*Log[x]))/(10*(b + a*x^(1/6)))

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Maple [A]
time = 0.07, size = 116, normalized size = 0.30

method	result	size
derivativedivides	\(\frac {\left (\frac {x^{\frac {1}{3}} a^{2}+2 a b \,x^{\frac {1}{6}}+b^{2}}{x^{\frac {1}{3}}}\right )^{\frac {7}{2}} x \left (10 a^{7} x^{\frac {7}{6}}+84 a^{6} b x +315 a^{5} b^{2} x^{\frac {5}{6}}+700 a^{4} b^{3} x^{\frac {2}{3}}+1050 a^{3} b^{4} \sqrt {x}+70 a \,b^{6} \ln \left (x \right ) x^{\frac {1}{6}}+1260 a^{2} b^{5} x^{\frac {1}{3}}-60 b^{7}\right )}{10 \left (a \,x^{\frac {1}{6}}+b \right )^{7}}\)	\(113\)
default	\(\frac {\sqrt {\frac {a^{2} \sqrt {x}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {1}{6}}}{\sqrt {x}}}\, \left (10 a^{7} x^{\frac {7}{6}}+84 a^{6} b x +315 a^{5} b^{2} x^{\frac {5}{6}}+700 a^{4} b^{3} x^{\frac {2}{3}}+1050 a^{3} b^{4} \sqrt {x}+70 a \,b^{6} \ln \left (x \right ) x^{\frac {1}{6}}+1260 a^{2} b^{5} x^{\frac {1}{3}}-60 b^{7}\right )}{10 a \,x^{\frac {1}{6}}+10 b}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/10*((a^2*x^(1/2)+2*a*b*x^(1/3)+b^2*x^(1/6))/x^(1/2))^(1/2)*(10*a^7*x^(7/6)+84*a^6*b*x+315*a^5*b^2*x^(5/6)+70
0*a^4*b^3*x^(2/3)+1050*a^3*b^4*x^(1/2)+70*a*b^6*ln(x)*x^(1/6)+1260*a^2*b^5*x^(1/3)-60*b^7)/(a*x^(1/6)+b)

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Maxima [A]
time = 0.30, size = 79, normalized size = 0.20 \begin {gather*} 7 \, a b^{6} \log \left (x\right ) + \frac {10 \, a^{7} x^{\frac {7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac {5}{6}} + 700 \, a^{4} b^{3} x^{\frac {2}{3}} + 1050 \, a^{3} b^{4} \sqrt {x} + 1260 \, a^{2} b^{5} x^{\frac {1}{3}} - 60 \, b^{7}}{10 \, x^{\frac {1}{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="maxima")

[Out]

7*a*b^6*log(x) + 1/10*(10*a^7*x^(7/6) + 84*a^6*b*x + 315*a^5*b^2*x^(5/6) + 700*a^4*b^3*x^(2/3) + 1050*a^3*b^4*
sqrt(x) + 1260*a^2*b^5*x^(1/3) - 60*b^7)/x^(1/6)

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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((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((a**2+b**2/x**(1/3)+2*a*b/x**(1/6))**(7/2),x)

[Out]

Timed out

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Giac [A]
time = 4.49, size = 172, normalized size = 0.44 \begin {gather*} a^{7} x \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {42}{5} \, a^{6} b x^{\frac {5}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {63}{2} \, a^{5} b^{2} x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt {x} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac {1}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) - \frac {6 \, b^{7} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right )}{x^{\frac {1}{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="giac")

[Out]

a^7*x*sgn(a*x + b*x^(5/6))*sgn(x) + 7*a*b^6*log(abs(x))*sgn(a*x + b*x^(5/6))*sgn(x) + 42/5*a^6*b*x^(5/6)*sgn(a
*x + b*x^(5/6))*sgn(x) + 63/2*a^5*b^2*x^(2/3)*sgn(a*x + b*x^(5/6))*sgn(x) + 70*a^4*b^3*sqrt(x)*sgn(a*x + b*x^(
5/6))*sgn(x) + 105*a^3*b^4*x^(1/3)*sgn(a*x + b*x^(5/6))*sgn(x) + 126*a^2*b^5*x^(1/6)*sgn(a*x + b*x^(5/6))*sgn(
x) - 6*b^7*sgn(a*x + b*x^(5/6))*sgn(x)/x^(1/6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a^2+\frac {b^2}{x^{1/3}}+\frac {2\,a\,b}{x^{1/6}}\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x)

[Out]

int((a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2), x)

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