3.1.35 \(\int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\) [35]

Optimal. Leaf size=1513 \[ \frac {3 c^4 d^2 \left (a+b x^3\right )^{2/3}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}-\frac {3 c^3 d^3 x \left (a+b x^3\right )^{2/3}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {4 b c^4 d^2 \left (a+b x^3\right )^{2/3}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c d^2 \left (b c^3-3 a d^3\right ) \left (a+b x^3\right )^{2/3}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^3 \left (3 b c^3-7 a d^3\right ) x \left (a+b x^3\right )^{2/3}}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {d^3 \left (9 b c^3-5 a d^3\right ) x \left (a+b x^3\right )^{2/3}}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {7 d^3 \left (3 b c^3+a d^3\right ) x \left (a+b x^3\right )^{2/3}}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {3 d x^2 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{3},3;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^4 \sqrt [3]{a+b x^3}}+\frac {6 d^4 x^5 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {5}{3};\frac {1}{3},3;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{5 c^7 \sqrt [3]{a+b x^3}}+\frac {2 a^2 d^6 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a d^3 \left (3 b c^3-a d^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b c^3 d^3+5 a^2 d^6\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {4 b^2 c^4 \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {2 b^2 c^4 \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{7/3}}+\frac {a^2 d^6 \log \left (c^3+d^3 x^3\right )}{27 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {b c \left (b c^3-3 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a d^3 \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b c^3 d^3+5 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {a^2 d^6 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{9 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 a d^3 \left (3 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {\left (9 b^2 c^6-12 a b c^3 d^3+5 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {2 b^2 c^4 \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{3 \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{6 \left (b c^3-a d^3\right )^{7/3}} \]

[Out]

3/2*c^4*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2-3/2*c^3*d^3*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x
^3+c^3)^2+4/3*b*c^4*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/3*c*d^2*(-3*a*d^3+b*c^3)*(b*x^3+a)^(2
/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/18*d^3*(-7*a*d^3+3*b*c^3)*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3
)-1/18*d^3*(-5*a*d^3+9*b*c^3)*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-7/18*d^3*(a*d^3+3*b*c^3)*x*(b*x
^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-3/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,3,5/3,-b*x^3/a,-d^3*x^
3/c^3)/c^4/(b*x^3+a)^(1/3)+6/5*d^4*x^5*(1+b*x^3/a)^(1/3)*AppellF1(5/3,1/3,3,8/3,-b*x^3/a,-d^3*x^3/c^3)/c^7/(b*
x^3+a)^(1/3)+2/9*b^2*c^4*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(7/3)+1/27*a^2*d^6*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^
(7/3)-1/18*b*c*(-3*a*d^3+b*c^3)*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(7/3)+7/54*a*d^3*(-a*d^3+3*b*c^3)*ln(d^3*x^3+c^
3)/c^2/(-a*d^3+b*c^3)^(7/3)+1/54*(5*a^2*d^6-12*a*b*c^3*d^3+9*b^2*c^6)*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(7/3)
-1/9*a^2*d^6*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(7/3)-7/18*a*d^3*(-a*d^3+3*b*c^3)
*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(7/3)-1/18*(5*a^2*d^6-12*a*b*c^3*d^3+9*b^2*c^
6)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(7/3)-2/3*b^2*c^4*ln((-a*d^3+b*c^3)^(1/3)+d
*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(7/3)+1/6*b*c*(-3*a*d^3+b*c^3)*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/(-a
*d^3+b*c^3)^(7/3)+2/27*a^2*d^6*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/c^2/(-a*d^3+
b*c^3)^(7/3)*3^(1/2)+7/27*a*d^3*(-a*d^3+3*b*c^3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(
1/2))/c^2/(-a*d^3+b*c^3)^(7/3)*3^(1/2)+1/27*(5*a^2*d^6-12*a*b*c^3*d^3+9*b^2*c^6)*arctan(1/3*(1+2*(-a*d^3+b*c^3
)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/c^2/(-a*d^3+b*c^3)^(7/3)*3^(1/2)-4/9*b^2*c^4*arctan(1/3*(1-2*d*(b*x^3+a)
^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/(-a*d^3+b*c^3)^(7/3)*3^(1/2)+1/9*b*c*(-3*a*d^3+b*c^3)*arctan(1/3*(1-2*d*
(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/(-a*d^3+b*c^3)^(7/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.32, antiderivative size = 1513, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 17, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {2181, 425, 541, 12, 384, 525, 524, 455, 44, 58, 631, 210, 31, 482, 457, 79, 481} \begin {gather*} \frac {2 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right ) d^6}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {a^2 \log \left (c^3+d^3 x^3\right ) d^6}{27 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {a^2 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^6}{9 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {6 x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},3;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d^4}{5 c^7 \sqrt [3]{b x^3+a}}+\frac {7 a \left (3 b c^3-a d^3\right ) \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right ) d^3}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right ) d^3}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^3}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 \left (3 b c^3+a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {\left (3 b c^3-7 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {\left (9 b c^3-5 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {3 c^3 x \left (b x^3+a\right )^{2/3} d^3}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {4 b c^4 \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c \left (b c^3-3 a d^3\right ) \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 \left (b x^3+a\right )^{2/3} d^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}-\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},3;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d}{2 c^4 \sqrt [3]{b x^3+a}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {4 b^2 c^4 \text {ArcTan}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \text {ArcTan}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {2 b^2 c^4 \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{7/3}}-\frac {b c \left (b c^3-3 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{7/3}}-\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {2 b^2 c^4 \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{3 \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{6 \left (b c^3-a d^3\right )^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c^4*d^2*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) - (3*c^3*d^3*x*(a + b*x^3)^(2/3))/(2*(b*c^
3 - a*d^3)*(c^3 + d^3*x^3)^2) + (4*b*c^4*d^2*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2
*(b*c^3 - 3*a*d^3)*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^3*(3*b*c^3 - 7*a*d^3)*x*(a +
b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (d^3*(9*b*c^3 - 5*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^
3 - a*d^3)^2*(c^3 + d^3*x^3)) - (7*d^3*(3*b*c^3 + a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3
*x^3)) - (3*d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 3, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2*c^4*(a
+ b*x^3)^(1/3)) + (6*d^4*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 3, 8/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])
/(5*c^7*(a + b*x^3)^(1/3)) + (2*a^2*d^6*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]
])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(7/3)) + (7*a*d^3*(3*b*c^3 - a*d^3)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/
(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*
d^6)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(
7/3)) - (4*b^2*c^4*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*(b*c^3 - a*
d^3)^(7/3)) + (b*c*(b*c^3 - 3*a*d^3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(3*S
qrt[3]*(b*c^3 - a*d^3)^(7/3)) + (2*b^2*c^4*Log[c^3 + d^3*x^3])/(9*(b*c^3 - a*d^3)^(7/3)) + (a^2*d^6*Log[c^3 +
d^3*x^3])/(27*c^2*(b*c^3 - a*d^3)^(7/3)) - (b*c*(b*c^3 - 3*a*d^3)*Log[c^3 + d^3*x^3])/(18*(b*c^3 - a*d^3)^(7/3
)) + (7*a*d^3*(3*b*c^3 - a*d^3)*Log[c^3 + d^3*x^3])/(54*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*
d^3 + 5*a^2*d^6)*Log[c^3 + d^3*x^3])/(54*c^2*(b*c^3 - a*d^3)^(7/3)) - (a^2*d^6*Log[((b*c^3 - a*d^3)^(1/3)*x)/c
- (a + b*x^3)^(1/3)])/(9*c^2*(b*c^3 - a*d^3)^(7/3)) - (7*a*d^3*(3*b*c^3 - a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x
)/c - (a + b*x^3)^(1/3)])/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*Log[((b*c
^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - (2*b^2*c^4*Log[(b*c^3 - a*d^3)^(
1/3) + d*(a + b*x^3)^(1/3)])/(3*(b*c^3 - a*d^3)^(7/3)) + (b*c*(b*c^3 - 3*a*d^3)*Log[(b*c^3 - a*d^3)^(1/3) + d*
(a + b*x^3)^(1/3)])/(6*(b*c^3 - a*d^3)^(7/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 457

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

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 482

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

Rule 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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])

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx &=\int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\\ \end {align*}

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Mathematica [F]
time = 20.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right )^{3} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((b*x^3 + a)^(1/3)*(d*x + 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/(d*x+c)^3/(b*x^3+a)^(1/3),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}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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