3.755 \(\int \frac {x^6}{(a+b x^3)^{4/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=260 \[ -\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{4/3} d}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {c^{4/3} \log \left (c+d x^3\right )}{6 d (b c-a d)^{4/3}}+\frac {c^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d (b c-a d)^{4/3}}-\frac {c^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d (b c-a d)^{4/3}}+\frac {a x}{b \sqrt [3]{a+b x^3} (b c-a d)} \]

[Out]

a*x/b/(-a*d+b*c)/(b*x^3+a)^(1/3)-1/6*c^(4/3)*ln(d*x^3+c)/d/(-a*d+b*c)^(4/3)+1/2*c^(4/3)*ln((-a*d+b*c)^(1/3)*x/
c^(1/3)-(b*x^3+a)^(1/3))/d/(-a*d+b*c)^(4/3)-1/2*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(4/3)/d+1/3*arctan(1/3*(1+2*b
^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(4/3)/d*3^(1/2)-1/3*c^(4/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b
*x^3+a)^(1/3))*3^(1/2))/d/(-a*d+b*c)^(4/3)*3^(1/2)

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Rubi [C]  time = 0.06, antiderivative size = 67, normalized size of antiderivative = 0.26, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac {x^7 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {7}{3};\frac {4}{3},1;\frac {10}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{7 a c \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

(x^7*(1 + (b*x^3)/a)^(1/3)*AppellF1[7/3, 4/3, 1, 10/3, -((b*x^3)/a), -((d*x^3)/c)])/(7*a*c*(a + b*x^3)^(1/3))

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {x^6}{\left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=\frac {x^7 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {7}{3};\frac {4}{3},1;\frac {10}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{7 a c \sqrt [3]{a+b x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.45, size = 309, normalized size = 1.19 \[ \frac {3 x^4 \sqrt [3]{\frac {b x^3}{a}+1} (b c-a d)^{4/3} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 a c \left (\sqrt [3]{c} \sqrt [3]{a+b x^3} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 \sqrt [3]{c} \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-6 x \sqrt [3]{b c-a d}\right )}{12 b c \sqrt [3]{a+b x^3} (b c-a d)^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

(3*(b*c - a*d)^(4/3)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*a*c*
(-6*(b*c - a*d)^(1/3)*x + 2*Sqrt[3]*c^(1/3)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b
+ a*x^3)^(1/3)))/Sqrt[3]] - 2*c^(1/3)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]
+ c^(1/3)*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1
/3)*x)/(b + a*x^3)^(1/3)]))/(12*b*c*(b*c - a*d)^(4/3)*(a + b*x^3)^(1/3))

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fricas [B]  time = 0.93, size = 1127, normalized size = 4.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

[1/6*(6*(b*x^3 + a)^(2/3)*a*b*d*x + 3*sqrt(1/3)*(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^3)*sqrt((-b)^(1/3)/b)
*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 +
2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 2*sqrt(3)*(b^3*c*x^3 + a*b^2*c)*(-c/(b*c - a*d))
^(1/3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-c/(b*c - a*d))^(1/3))/x) - 2*((b^2*c - a*b*d)*x^
3 + a*b*c - a^2*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + ((b^2*c - a*b*d)*x^3 + a*b*c - a^2*d
)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) - 2*(b^3*c*x^3 + a
*b^2*c)*(-c/(b*c - a*d))^(1/3)*log(-((b*c - a*d)*x*(-c/(b*c - a*d))^(2/3) - (b*x^3 + a)^(1/3)*c)/x) + (b^3*c*x
^3 + a*b^2*c)*(-c/(b*c - a*d))^(1/3)*log(-((b*c - a*d)*x^2*(-c/(b*c - a*d))^(1/3) - (b*x^3 + a)^(1/3)*(b*c - a
*d)*x*(-c/(b*c - a*d))^(2/3) - (b*x^3 + a)^(2/3)*c)/x^2))/(a*b^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2)*x^3
), 1/6*(6*(b*x^3 + a)^(2/3)*a*b*d*x - 6*sqrt(1/3)*(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^3)*sqrt(-(-b)^(1/3)
/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 2*sqrt(3)*(b^3*c*x^3 + a*b
^2*c)*(-c/(b*c - a*d))^(1/3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-c/(b*c - a*d))^(1/3))/x) -
2*((b^2*c - a*b*d)*x^3 + a*b*c - a^2*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + ((b^2*c - a*b*
d)*x^3 + a*b*c - a^2*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x
^2) - 2*(b^3*c*x^3 + a*b^2*c)*(-c/(b*c - a*d))^(1/3)*log(-((b*c - a*d)*x*(-c/(b*c - a*d))^(2/3) - (b*x^3 + a)^
(1/3)*c)/x) + (b^3*c*x^3 + a*b^2*c)*(-c/(b*c - a*d))^(1/3)*log(-((b*c - a*d)*x^2*(-c/(b*c - a*d))^(1/3) - (b*x
^3 + a)^(1/3)*(b*c - a*d)*x*(-c/(b*c - a*d))^(2/3) - (b*x^3 + a)^(2/3)*c)/x^2))/(a*b^3*c*d - a^2*b^2*d^2 + (b^
4*c*d - a*b^3*d^2)*x^3)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(x^6/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)

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maple [F]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(x^6/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^6}{{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x)

[Out]

int(x^6/((a + b*x^3)^(4/3)*(c + d*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**3+a)**(4/3)/(d*x**3+c),x)

[Out]

Integral(x**6/((a + b*x**3)**(4/3)*(c + d*x**3)), x)

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