3.2.21 \(\int \frac {1}{(a+b x^3)^{8/3} (c+d x^3)^3} \, dx\) [121]

Optimal. Leaf size=62 \[ \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \]

[Out]

x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,8/3,3,4/3,-b*x^3/a,-d*x^3/c)/a^2/c^3/(b*x^3+a)^(2/3)

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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440} \begin {gather*} \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^(8/3)*(c + d*x^3)^3),x]

[Out]

(x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 8/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(a^2*c^3*(a + b*x^3)^(2/3))

Rule 440

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

Rule 441

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^3} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{8/3} \left (c+d x^3\right )^3} \, dx}{a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(515\) vs. \(2(62)=124\).
time = 11.08, size = 515, normalized size = 8.31 \begin {gather*} \frac {x \left (b d \left (36 b^3 c^3-171 a b^2 c^2 d-110 a^2 b c d^2+25 a^3 d^3\right ) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (\frac {36 b^5 c^3 x^3 \left (c+d x^3\right )^2+9 a b^4 c^2 \left (6 c-19 d x^3\right ) \left (c+d x^3\right )^2+5 a^5 d^4 \left (8 c+5 d x^3\right )+5 a^3 b^2 d^3 x^3 \left (-50 c^2-36 c d x^3+5 d^2 x^6\right )+5 a^4 b d^3 \left (-25 c^2-6 c d x^3+10 d^2 x^6\right )-a^2 b^3 c d \left (189 c^3+378 c^2 d x^3+314 c d^2 x^6+110 d^3 x^9\right )}{a+b x^3}+\frac {4 a c \left (36 b^4 c^4-171 a b^3 c^3 d+540 a^2 b^2 c^2 d^2-235 a^3 b c d^3+50 a^4 d^4\right ) \left (c+d x^3\right ) F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{\left (c+d x^3\right )^2}\right )}{360 a^2 c^3 (b c-a d)^4 \left (a+b x^3\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(8/3)*(c + d*x^3)^3),x]

[Out]

(x*(b*d*(36*b^3*c^3 - 171*a*b^2*c^2*d - 110*a^2*b*c*d^2 + 25*a^3*d^3)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3,
2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (4*c*((36*b^5*c^3*x^3*(c + d*x^3)^2 + 9*a*b^4*c^2*(6*c - 19*d*x^3)*
(c + d*x^3)^2 + 5*a^5*d^4*(8*c + 5*d*x^3) + 5*a^3*b^2*d^3*x^3*(-50*c^2 - 36*c*d*x^3 + 5*d^2*x^6) + 5*a^4*b*d^3
*(-25*c^2 - 6*c*d*x^3 + 10*d^2*x^6) - a^2*b^3*c*d*(189*c^3 + 378*c^2*d*x^3 + 314*c*d^2*x^6 + 110*d^3*x^9))/(a
+ b*x^3) + (4*a*c*(36*b^4*c^4 - 171*a*b^3*c^3*d + 540*a^2*b^2*c^2*d^2 - 235*a^3*b*c*d^3 + 50*a^4*d^4)*(c + d*x
^3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -(
(d*x^3)/c)] - x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1,
7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(c + d*x^3)^2))/(360*a^2*c^3*(b*c - a*d)^4*(a + b*x^3)^(2/3))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {8}{3}} \left (d \,x^{3}+c \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(8/3)/(d*x^3+c)^3,x)

[Out]

int(1/(b*x^3+a)^(8/3)/(d*x^3+c)^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(8/3)/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(8/3)*(d*x^3 + c)^3), x)

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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/(b*x^3+a)^(8/3)/(d*x^3+c)^3,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(1/(b*x**3+a)**(8/3)/(d*x**3+c)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(8/3)/(d*x^3+c)^3,x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(8/3)*(d*x^3 + c)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (b\,x^3+a\right )}^{8/3}\,{\left (d\,x^3+c\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^3)^(8/3)*(c + d*x^3)^3),x)

[Out]

int(1/((a + b*x^3)^(8/3)*(c + d*x^3)^3), x)

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