3.103 \(\int \frac {1}{(a+b x^3)^{7/3} (c+d x^3)^2} \, dx\)

Optimal. Leaf size=324 \[ \frac {b x \left (-4 a^2 d^2-33 a b c d+9 b^2 c^2\right )}{12 a^2 c \sqrt [3]{a+b x^3} (b c-a d)^3}+\frac {d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac {d^2 (9 b c-2 a d) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}}+\frac {d^2 (9 b c-2 a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{10/3}}-\frac {d x}{3 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}+\frac {b x (4 a d+3 b c)}{12 a c \left (a+b x^3\right )^{4/3} (b c-a d)^2} \]

[Out]

1/12*b*(4*a*d+3*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^3+a)^(4/3)+1/12*b*(-4*a^2*d^2-33*a*b*c*d+9*b^2*c^2)*x/a^2/c/(-a*d
+b*c)^3/(b*x^3+a)^(1/3)-1/3*d*x/c/(-a*d+b*c)/(b*x^3+a)^(4/3)/(d*x^3+c)+1/18*d^2*(-2*a*d+9*b*c)*ln(d*x^3+c)/c^(
5/3)/(-a*d+b*c)^(10/3)-1/6*d^2*(-2*a*d+9*b*c)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(5/3)/(-a*d+b*c
)^(10/3)+1/9*d^2*(-2*a*d+9*b*c)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(5/3)/(
-a*d+b*c)^(10/3)*3^(1/2)

________________________________________________________________________________________

Rubi [C]  time = 5.69, antiderivative size = 1214, normalized size of antiderivative = 3.75, 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 = {430, 429} \[ \frac {6804 d^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}+1134 d^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}+21546 c d^2 (b c-a d)^4 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}+3402 c d^2 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}+26325 c^2 d^3 (b c-a d)^2 \left (b x^3+a\right )^2 x^{15}+22680 c^2 d (b c-a d)^4 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+3402 c^2 d (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+589680 c^3 d^3 (b c-a d) \left (b x^3+a\right )^3 x^{12}+84240 c^3 d^2 (b c-a d)^2 \left (b x^3+a\right )^2 x^{12}-966420 c^3 d^3 (b c-a d) \left (b x^3+a\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+7938 c^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+1134 c^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-1506960 c^4 d^3 \left (b x^3+a\right )^4 x^9+1916460 c^4 d^2 (b c-a d) \left (b x^3+a\right )^3 x^9+89505 c^4 d (b c-a d)^2 \left (b x^3+a\right )^2 x^9+1506960 c^4 d^3 \left (b x^3+a\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-3144960 c^4 d^2 (b c-a d) \left (b x^3+a\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-4914000 c^5 d^2 \left (b x^3+a\right )^4 x^6+2113020 c^5 d (b c-a d) \left (b x^3+a\right )^3 x^6+26130 c^5 (b c-a d)^2 \left (b x^3+a\right )^2 x^6+4914000 c^5 d^2 \left (b x^3+a\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-3478020 c^5 d (b c-a d) \left (b x^3+a\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-5460000 c^6 d \left (b x^3+a\right )^4 x^3+748020 c^6 (b c-a d) \left (b x^3+a\right )^3 x^3+5460000 c^6 d \left (b x^3+a\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-1248520 c^6 (b c-a d) \left (b x^3+a\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-2002000 c^7 \left (b x^3+a\right )^4+2002000 c^7 \left (b x^3+a\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{21840 c^5 (b c-a d)^3 x^8 \left (b x^3+a\right )^{10/3} \left (d x^3+c\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(26130*c^5*(b*c - a*d)^2*x^6*(a + b*x^3)^2 + 89505*c^4*d*(b*c - a*d)^2*x^9*(a + b*x^3)^2 + 84240*c^3*d^2*(b*c
- a*d)^2*x^12*(a + b*x^3)^2 + 26325*c^2*d^3*(b*c - a*d)^2*x^15*(a + b*x^3)^2 + 748020*c^6*(b*c - a*d)*x^3*(a +
b*x^3)^3 + 2113020*c^5*d*(b*c - a*d)*x^6*(a + b*x^3)^3 + 1916460*c^4*d^2*(b*c - a*d)*x^9*(a + b*x^3)^3 + 5896
80*c^3*d^3*(b*c - a*d)*x^12*(a + b*x^3)^3 - 2002000*c^7*(a + b*x^3)^4 - 5460000*c^6*d*x^3*(a + b*x^3)^4 - 4914
000*c^5*d^2*x^6*(a + b*x^3)^4 - 1506960*c^4*d^3*x^9*(a + b*x^3)^4 - 1248520*c^6*(b*c - a*d)*x^3*(a + b*x^3)^3*
Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 3478020*c^5*d*(b*c - a*d)*x^6*(a + b*x^3)^
3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 3144960*c^4*d^2*(b*c - a*d)*x^9*(a + b*x
^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 966420*c^3*d^3*(b*c - a*d)*x^12*(a +
b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 2002000*c^7*(a + b*x^3)^4*Hyperg
eometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 5460000*c^6*d*x^3*(a + b*x^3)^4*Hypergeometric2F
1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 4914000*c^5*d^2*x^6*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1
, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1506960*c^4*d^3*x^9*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, (
(b*c - a*d)*x^3)/(c*(a + b*x^3))] + 7938*c^3*(b*c - a*d)^4*x^12*HypergeometricPFQ[{2, 2, 10/3}, {1, 16/3}, ((b
*c - a*d)*x^3)/(c*(a + b*x^3))] + 22680*c^2*d*(b*c - a*d)^4*x^15*HypergeometricPFQ[{2, 2, 10/3}, {1, 16/3}, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))] + 21546*c*d^2*(b*c - a*d)^4*x^18*HypergeometricPFQ[{2, 2, 10/3}, {1, 16/3}, (
(b*c - a*d)*x^3)/(c*(a + b*x^3))] + 6804*d^3*(b*c - a*d)^4*x^21*HypergeometricPFQ[{2, 2, 10/3}, {1, 16/3}, ((b
*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c^3*(b*c - a*d)^4*x^12*HypergeometricPFQ[{2, 2, 2, 10/3}, {1, 1, 16/3},
((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3402*c^2*d*(b*c - a*d)^4*x^15*HypergeometricPFQ[{2, 2, 2, 10/3}, {1, 1,
16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3402*c*d^2*(b*c - a*d)^4*x^18*HypergeometricPFQ[{2, 2, 2, 10/3}, {
1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*d^3*(b*c - a*d)^4*x^21*HypergeometricPFQ[{2, 2, 2, 10/3
}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(21840*c^5*(b*c - a*d)^3*x^8*(a + b*x^3)^(10/3)*(c + d*x^
3))

Rule 429

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 430

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 )^{7/3} \left (c+d x^3\right )^2} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{7/3} \left (c+d x^3\right )^2} \, dx}{a^2 \sqrt [3]{a+b x^3}}\\ &=\frac {26130 c^5 (b c-a d)^2 x^6 \left (a+b x^3\right )^2+89505 c^4 d (b c-a d)^2 x^9 \left (a+b x^3\right )^2+84240 c^3 d^2 (b c-a d)^2 x^{12} \left (a+b x^3\right )^2+26325 c^2 d^3 (b c-a d)^2 x^{15} \left (a+b x^3\right )^2+748020 c^6 (b c-a d) x^3 \left (a+b x^3\right )^3+2113020 c^5 d (b c-a d) x^6 \left (a+b x^3\right )^3+1916460 c^4 d^2 (b c-a d) x^9 \left (a+b x^3\right )^3+589680 c^3 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^3-2002000 c^7 \left (a+b x^3\right )^4-5460000 c^6 d x^3 \left (a+b x^3\right )^4-4914000 c^5 d^2 x^6 \left (a+b x^3\right )^4-1506960 c^4 d^3 x^9 \left (a+b x^3\right )^4-1248520 c^6 (b c-a d) x^3 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-3478020 c^5 d (b c-a d) x^6 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-3144960 c^4 d^2 (b c-a d) x^9 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-966420 c^3 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+2002000 c^7 \left (a+b x^3\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+5460000 c^6 d x^3 \left (a+b x^3\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+4914000 c^5 d^2 x^6 \left (a+b x^3\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1506960 c^4 d^3 x^9 \left (a+b x^3\right )^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+7938 c^3 (b c-a d)^4 x^{12} \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+22680 c^2 d (b c-a d)^4 x^{15} \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+21546 c d^2 (b c-a d)^4 x^{18} \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+6804 d^3 (b c-a d)^4 x^{21} \, _3F_2\left (2,2,\frac {10}{3};1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1134 c^3 (b c-a d)^4 x^{12} \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+3402 c^2 d (b c-a d)^4 x^{15} \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+3402 c d^2 (b c-a d)^4 x^{18} \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1134 d^3 (b c-a d)^4 x^{21} \, _4F_3\left (2,2,2,\frac {10}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{21840 c^5 (b c-a d)^3 x^8 \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}\\ \end {align*}

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Mathematica [A]  time = 5.94, size = 288, normalized size = 0.89 \[ \frac {1}{36} \left (3 x \left (a+b x^3\right )^{2/3} \left (\frac {3 b^2 (11 a d-3 b c)}{a^2 \left (a+b x^3\right ) (a d-b c)^3}+\frac {3 b^2}{a \left (a+b x^3\right )^2 (b c-a d)^2}-\frac {4 d^3}{c \left (c+d x^3\right ) (b c-a d)^3}\right )+\frac {2 d^2 (9 b c-2 a d) \left (\log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )\right )}{c^{5/3} (b c-a d)^{10/3}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*x*(a + b*x^3)^(2/3)*((3*b^2)/(a*(b*c - a*d)^2*(a + b*x^3)^2) + (3*b^2*(-3*b*c + 11*a*d))/(a^2*(-(b*c) + a*d
)^3*(a + b*x^3)) - (4*d^3)/(c*(b*c - a*d)^3*(c + d*x^3))) + (2*d^2*(9*b*c - 2*a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*(
b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*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)]))/(c^(5/3)*(b*c - a*d)^(10/3)))/36

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)^2), x)

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maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {7}{3}} \left (d \,x^{3}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {7}{3}} \left (c + d x^{3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a)**(7/3)/(d*x**3+c)**2,x)

[Out]

Integral(1/((a + b*x**3)**(7/3)*(c + d*x**3)**2), x)

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