3.109 \(\int \frac{(a+b x^3)^{14/3}}{(c+d x^3)^3} \, dx\)

Optimal. Leaf size=541 \[ \frac{b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{18 c^2 d^3}-\frac{b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{18 c^2 d^4}-\frac{(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac{(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac{b^{8/3} \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 d^5}+\frac{b^{8/3} \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} d^5}-\frac{(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} d^5}-\frac{x \left (a+b x^3\right )^{8/3} (b c-a d) (5 a d+12 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

[Out]

-(b*(2*b*c - a*d)*(18*b^2*c^2 - 18*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/(18*c^2*d^4) + (b*(18*b^2*c^2 - 1
0*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(5/3))/(18*c^2*d^3) - ((b*c - a*d)*x*(a + b*x^3)^(11/3))/(6*c*d*(c + d*x^
3)^2) - ((b*c - a*d)*(12*b*c + 5*a*d)*x*(a + b*x^3)^(8/3))/(18*c^2*d^2*(c + d*x^3)) + (b^(8/3)*(54*b^2*c^2 - 1
26*a*b*c*d + 77*a^2*d^2)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*d^5) - ((b*c - a*d)
^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/
Sqrt[3]])/(9*Sqrt[3]*c^(8/3)*d^5) - ((b*c - a*d)^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*Log[c + d*x^3])/(
54*c^(8/3)*d^5) + ((b*c - a*d)^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) -
(a + b*x^3)^(1/3)])/(18*c^(8/3)*d^5) - (b^(8/3)*(54*b^2*c^2 - 126*a*b*c*d + 77*a^2*d^2)*Log[-(b^(1/3)*x) + (a
+ b*x^3)^(1/3)])/(18*d^5)

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Rubi [C]  time = 0.0273345, antiderivative size = 62, normalized size of antiderivative = 0.11, 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} \[ \frac{a^4 x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{14}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^3 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(14/3)/(c + d*x^3)^3,x]

[Out]

(a^4*x*(a + b*x^3)^(2/3)*AppellF1[1/3, -14/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c^3*(1 + (b*x^3)/a)^(2/3))

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])

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])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx &=\frac{\left (a^4 \left (a+b x^3\right )^{2/3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{a^4 x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{14}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^3 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 2.1929, size = 1171, normalized size = 2.16 \[ \frac{1}{108} \left (\frac{10 \left (2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac{(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^5}{c^{8/3} \sqrt [3]{b c-a d}}+\frac{6 b \left (2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac{(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^4}{c^{5/3} d \sqrt [3]{b c-a d}}+\frac{30 b^2 \left (2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac{(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^3}{c^{2/3} d^2 \sqrt [3]{b c-a d}}+\frac{231 b^3 x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) a^2}{c d^2 \sqrt [3]{b x^3+a}}-\frac{72 b^3 \sqrt [3]{c} \left (2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac{(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a^2}{d^3 \sqrt [3]{b c-a d}}-\frac{378 b^4 x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) a}{d^3 \sqrt [3]{b x^3+a}}+\frac{36 b^4 c^{4/3} \left (2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}\right )+\log \left (\frac{(b c-a d)^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )\right ) a}{d^4 \sqrt [3]{b c-a d}}+\frac{6 x \left (b x^3+a\right )^{2/3} \left (3 d x^3 b^4-2 (9 b c-13 a d) b^3-\frac{(b c-a d)^3 (21 b c+5 a d)}{c^2 \left (d x^3+c\right )}+\frac{3 (b c-a d)^4}{c \left (d x^3+c\right )^2}\right )}{d^4}+\frac{162 b^5 c x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{d^4 \sqrt [3]{b x^3+a}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(14/3)/(c + d*x^3)^3,x]

[Out]

((6*x*(a + b*x^3)^(2/3)*(-2*b^3*(9*b*c - 13*a*d) + 3*b^4*d*x^3 + (3*(b*c - a*d)^4)/(c*(c + d*x^3)^2) - ((b*c -
a*d)^3*(21*b*c + 5*a*d))/(c^2*(c + d*x^3))))/d^4 + (162*b^5*c*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)])/(d^4*(a + b*x^3)^(1/3)) - (378*a*b^4*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)])/(d^3*(a + b*x^3)^(1/3)) + (231*a^2*b^3*x^4*(1 + (b*x^3)/a)^(1/3)*A
ppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*d^2*(a + b*x^3)^(1/3)) + (10*a^5*(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^(8/3)*(b*c - a*d)^(1/3)) + (36*a*b^4*c^(4/3)*(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)]))/(d^4*(b*c -
a*d)^(1/3)) - (72*a^2*b^3*c^(1/3)*(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)]))/(d^3*(b*c - a*d)^(1/3)) + (30*a^3*b^2*(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^(2/3)*d^2*(b*c - a*d)^(1/3)) + (6*a^4*b*(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)*d*(b*c - a*d)^(1/3)))/108

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Maple [F]  time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{14}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(14/3)/(d*x^3+c)^3,x)

[Out]

int((b*x^3+a)^(14/3)/(d*x^3+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{14}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(14/3)/(d*x^3 + c)^3, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(14/3)/(d*x**3+c)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{14}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(14/3)/(d*x^3 + c)^3, x)