3.754 \(\int \frac{x^9}{(a+b x^3)^{4/3} (c+d x^3)} \, dx\)
Optimal. Leaf size=322 \[ \frac{(4 a d+3 b c) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{7/3} d^2}-\frac{(4 a d+3 b c) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{7/3} d^2}+\frac{x \left (a+b x^3\right )^{2/3} (b c-4 a d)}{3 b^2 d (b c-a d)}+\frac{c^{7/3} \log \left (c+d x^3\right )}{6 d^2 (b c-a d)^{4/3}}-\frac{c^{7/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2 (b c-a d)^{4/3}}+\frac{c^{7/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2 (b c-a d)^{4/3}}+\frac{a x^4}{b \sqrt [3]{a+b x^3} (b c-a d)} \]
[Out]
(a*x^4)/(b*(b*c - a*d)*(a + b*x^3)^(1/3)) + ((b*c - 4*a*d)*x*(a + b*x^3)^(2/3))/(3*b^2*d*(b*c - a*d)) - ((3*b*
c + 4*a*d)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(7/3)*d^2) + (c^(7/3)*ArcTan[(1
+ (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*d^2*(b*c - a*d)^(4/3)) + (c^(7/3)*L
og[c + d*x^3])/(6*d^2*(b*c - a*d)^(4/3)) - (c^(7/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2
*d^2*(b*c - a*d)^(4/3)) + ((3*b*c + 4*a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(6*b^(7/3)*d^2)
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Rubi [C] time = 0.0603097, antiderivative size = 67, normalized size of antiderivative =
0.21, 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} \[ \frac{x^{10} \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{10}{3};\frac{4}{3},1;\frac{13}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{10 a c \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
Int[x^9/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
[Out]
(x^10*(1 + (b*x^3)/a)^(1/3)*AppellF1[10/3, 4/3, 1, 13/3, -((b*x^3)/a), -((d*x^3)/c)])/(10*a*c*(a + b*x^3)^(1/3
))
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])
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])
Rubi steps
\begin{align*} \int \frac{x^9}{\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^9}{\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^{10} \sqrt [3]{1+\frac{b x^3}{a}} F_1\left (\frac{10}{3};\frac{4}{3},1;\frac{13}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{10 a c \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.828278, size = 504, normalized size = 1.57 \[ -\frac{3 x^4 \sqrt [3]{\frac{b x^3}{a}+1} \sqrt [3]{b c-a d} \left (-4 a^2 d^2+a b c d+3 b^2 c^2\right ) F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 c \left (-4 a^2 \sqrt [3]{c} d \sqrt [3]{a+b x^3} \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )+24 a^2 d x \sqrt [3]{b c-a d}-6 b^2 c x^4 \sqrt [3]{b c-a d}+a b c^{4/3} \sqrt [3]{a+b x^3} \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )+6 a b d x^4 \sqrt [3]{b c-a d}+2 a \sqrt [3]{c} \sqrt [3]{a+b x^3} (4 a d-b c) \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )-2 \sqrt{3} a \sqrt [3]{c} \sqrt [3]{a+b x^3} (4 a d-b c) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )-6 a b c x \sqrt [3]{b c-a d}\right )}{36 b^2 c d \sqrt [3]{a+b x^3} (b c-a d)^{4/3}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x^9/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
[Out]
-(3*(b*c - a*d)^(1/3)*(3*b^2*c^2 + a*b*c*d - 4*a^2*d^2)*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*c*(-6*a*b*c*(b*c - a*d)^(1/3)*x + 24*a^2*d*(b*c - a*d)^(1/3)*x - 6*b^2*c*(b*c -
a*d)^(1/3)*x^4 + 6*a*b*d*(b*c - a*d)^(1/3)*x^4 - 2*Sqrt[3]*a*c^(1/3)*(-(b*c) + 4*a*d)*(a + b*x^3)^(1/3)*ArcTa
n[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] + 2*a*c^(1/3)*(-(b*c) + 4*a*d)*(a + b*x^3
)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + a*b*c^(4/3)*(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)] - 4*a^2*c^(1/3)*d*
(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)]))/(36*b^2*c*d*(b*c - a*d)^(4/3)*(a + b*x^3)^(1/3))
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{9}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^9/(b*x^3+a)^(4/3)/(d*x^3+c),x)
[Out]
int(x^9/(b*x^3+a)^(4/3)/(d*x^3+c),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^9/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")
[Out]
integrate(x^9/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)
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Fricas [B] time = 6.05854, size = 2989, normalized size = 9.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^9/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")
[Out]
[1/18*(3*sqrt(1/3)*(3*a*b^3*c^2 + a^2*b^2*c*d - 4*a^3*b*d^2 + (3*b^4*c^2 + a*b^3*c*d - 4*a^2*b^2*d^2)*x^3)*sqr
t(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*(b^(4/3)*x^3 + (b*x^3 + a)^(1/3)*b*x
^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) - 6*sqrt(3)*(b^4*c^2*x^3 + a*b^3*c^2)*(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*(3*a*b^2*c^2 + a^2
*b*c*d - 4*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x^3)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x
) - (3*a*b^2*c^2 + a^2*b*c*d - 4*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x^3)*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) - 6*(b^4*c^2*x^3 + a*b^3*c^2)*(c/(b*c - a*d))^(1/3)*l
og(-((b*c - a*d)*x*(c/(b*c - a*d))^(2/3) - (b*x^3 + a)^(1/3)*c)/x) + 3*(b^4*c^2*x^3 + a*b^3*c^2)*(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) + 6*((b^3*c*d - a*b^2*d^2)*x^4 + (a*b^2*c*d - 4*a^2*b*d^2)*x)*(b*x^3 + a)^(2/3))/(a*
b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^3), -1/18*(6*sqrt(3)*(b^4*c^2*x^3 + a*b^3*c^2)*(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*(3*a*b^2*c^2 + a^
2*b*c*d - 4*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x^3)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/
x) + (3*a*b^2*c^2 + a^2*b*c*d - 4*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x^3)*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) + 6*(b^4*c^2*x^3 + a*b^3*c^2)*(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) - 3*(b^4*c^2*x^3 + a*b^3*c^2)*(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) - 6*sqrt(1/3)*(3*a*b^3*c^2 + a^2*b^2*c*d - 4*a^3*b*d^2 + (3*b^4*c^2 + a*b^3*c*d - 4
*a^2*b^2*d^2)*x^3)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3) - 6*((b^3*c*d - a*b
^2*d^2)*x^4 + (a*b^2*c*d - 4*a^2*b*d^2)*x)*(b*x^3 + a)^(2/3))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*
d^3)*x^3)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{\left (a + b x^{3}\right )^{\frac{4}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**9/(b*x**3+a)**(4/3)/(d*x**3+c),x)
[Out]
Integral(x**9/((a + b*x**3)**(4/3)*(c + d*x**3)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^9/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")
[Out]
integrate(x^9/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)