3.56 \(\int \frac{x^6 \left (A+B x^3\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=183 \[ -\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}-\frac{a x (A b-a B)}{b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^7}{7 b} \]

[Out]

-((a*(A*b - a*B)*x)/b^3) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^7)/(7*b) - (a^(4/3)*
(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(10/3)
) + (a^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) - (a^(4/3)*(A*b
- a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3))

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Rubi [A]  time = 0.381213, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}-\frac{a x (A b-a B)}{b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x^3))/(a + b*x^3),x]

[Out]

-((a*(A*b - a*B)*x)/b^3) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^7)/(7*b) - (a^(4/3)*
(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(10/3)
) + (a^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) - (a^(4/3)*(A*b
- a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{7}}{7 b} + \frac{a^{\frac{4}{3}} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{10}{3}}} - \frac{a^{\frac{4}{3}} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{10}{3}}} - \frac{\sqrt{3} a^{\frac{4}{3}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{10}{3}}} + \frac{x^{4} \left (A b - B a\right )}{4 b^{2}} - \frac{\left (A b - B a\right ) \int a\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(B*x**3+A)/(b*x**3+a),x)

[Out]

B*x**7/(7*b) + a**(4/3)*(A*b - B*a)*log(a**(1/3) + b**(1/3)*x)/(3*b**(10/3)) - a
**(4/3)*(A*b - B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(1
0/3)) - sqrt(3)*a**(4/3)*(A*b - B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/
a**(1/3))/(3*b**(10/3)) + x**4*(A*b - B*a)/(4*b**2) - (A*b - B*a)*Integral(a, x)
/b**3

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Mathematica [A]  time = 0.27454, size = 171, normalized size = 0.93 \[ \frac{14 a^{4/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a^{4/3} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} a^{4/3} (a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+21 b^{4/3} x^4 (A b-a B)+84 a \sqrt [3]{b} x (a B-A b)+12 b^{7/3} B x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x^3))/(a + b*x^3),x]

[Out]

(84*a*b^(1/3)*(-(A*b) + a*B)*x + 21*b^(4/3)*(A*b - a*B)*x^4 + 12*b^(7/3)*B*x^7 +
28*Sqrt[3]*a^(4/3)*(-(A*b) + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] -
28*a^(4/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-(A*b) + a*B)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(84*b^(10/3))

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Maple [A]  time = 0.005, size = 249, normalized size = 1.4 \[{\frac{B{x}^{7}}{7\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{Bx{a}^{2}}{{b}^{3}}}+{\frac{A{a}^{2}}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B{a}^{3}}{3\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A{a}^{2}}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B{a}^{3}}{6\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}A}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{3}\sqrt{3}B}{3\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(B*x^3+A)/(b*x^3+a),x)

[Out]

1/7*B*x^7/b+1/4/b*A*x^4-1/4/b^2*B*x^4*a-1/b^2*A*x*a+1/b^3*B*x*a^2+1/3*a^2/b^3/(a
/b)^(2/3)*ln(x+(a/b)^(1/3))*A-1/3*a^3/b^4/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*B-1/6*a^
2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*A+1/6*a^3/b^4/(a/b)^(2/3)*ln
(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*B+1/3*a^2/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))*A-1/3*a^3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^6/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245927, size = 244, normalized size = 1.33 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (B a^{2} - A a b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3}{\left (B a^{2} - A a b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (B a^{2} - A a b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (4 \, B b^{2} x^{7} - 7 \,{\left (B a b - A b^{2}\right )} x^{4} + 28 \,{\left (B a^{2} - A a b\right )} x\right )}\right )}}{252 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^6/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/252*sqrt(3)*(14*sqrt(3)*(B*a^2 - A*a*b)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) +
(a/b)^(2/3)) - 28*sqrt(3)*(B*a^2 - A*a*b)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 84*
(B*a^2 - A*a*b)*(a/b)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3)*(a/b)^(1/3))/(a/b
)^(1/3)) + 3*sqrt(3)*(4*B*b^2*x^7 - 7*(B*a*b - A*b^2)*x^4 + 28*(B*a^2 - A*a*b)*x
))/b^3

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Sympy [A]  time = 2.45915, size = 110, normalized size = 0.6 \[ \frac{B x^{7}}{7 b} + \operatorname{RootSum}{\left (27 t^{3} b^{10} - A^{3} a^{4} b^{3} + 3 A^{2} B a^{5} b^{2} - 3 A B^{2} a^{6} b + B^{3} a^{7}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{3}}{- A a b + B a^{2}} + x \right )} \right )\right )} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(B*x**3+A)/(b*x**3+a),x)

[Out]

B*x**7/(7*b) + RootSum(27*_t**3*b**10 - A**3*a**4*b**3 + 3*A**2*B*a**5*b**2 - 3*
A*B**2*a**6*b + B**3*a**7, Lambda(_t, _t*log(-3*_t*b**3/(-A*a*b + B*a**2) + x)))
- x**4*(-A*b + B*a)/(4*b**2) + x*(-A*a*b + B*a**2)/b**3

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GIAC/XCAS [A]  time = 0.219523, size = 293, normalized size = 1.6 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} A a b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} A a b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{{\left (B a^{3} b^{4} - A a^{2} b^{5}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac{4 \, B b^{6} x^{7} - 7 \, B a b^{5} x^{4} + 7 \, A b^{6} x^{4} + 28 \, B a^{2} b^{4} x - 28 \, A a b^{5} x}{28 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^6/(b*x^3 + a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*B*a^2 - (-a*b^2)^(1/3)*A*a*b)*arctan(1/3*sqrt(3)*(2
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^(1/3)*B*a^2 - (-a*b^2)^(1/3
)*A*a*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 1/3*(B*a^3*b^4 - A*a^2*b^
5)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/28*(4*B*b^6*x^7 - 7*B*a*b^
5*x^4 + 7*A*b^6*x^4 + 28*B*a^2*b^4*x - 28*A*a*b^5*x)/b^7