3.126 \(\int x^m \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{m+4} (a B+A b)}{m+4}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+7}}{m+7} \]

[Out]

(a*A*x^(1 + m))/(1 + m) + ((A*b + a*B)*x^(4 + m))/(4 + m) + (b*B*x^(7 + m))/(7 +
m)

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Rubi [A]  time = 0.0703345, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{x^{m+4} (a B+A b)}{m+4}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^(1 + m))/(1 + m) + ((A*b + a*B)*x^(4 + m))/(4 + m) + (b*B*x^(7 + m))/(7 +
m)

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Rubi in Sympy [A]  time = 8.51197, size = 37, normalized size = 0.82 \[ \frac{A a x^{m + 1}}{m + 1} + \frac{B b x^{m + 7}}{m + 7} + \frac{x^{m + 4} \left (A b + B a\right )}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**(m + 1)/(m + 1) + B*b*x**(m + 7)/(m + 7) + x**(m + 4)*(A*b + B*a)/(m + 4)

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Mathematica [A]  time = 0.0478, size = 41, normalized size = 0.91 \[ x^m \left (\frac{x^4 (a B+A b)}{m+4}+\frac{a A x}{m+1}+\frac{b B x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x^m*((a*A*x)/(1 + m) + ((A*b + a*B)*x^4)/(4 + m) + (b*B*x^7)/(7 + m))

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Maple [B]  time = 0.006, size = 110, normalized size = 2.4 \[{\frac{{x}^{1+m} \left ( Bb{m}^{2}{x}^{6}+5\,Bbm{x}^{6}+4\,bB{x}^{6}+Ab{m}^{2}{x}^{3}+Ba{m}^{2}{x}^{3}+8\,Abm{x}^{3}+8\,Bam{x}^{3}+7\,A{x}^{3}b+7\,B{x}^{3}a+Aa{m}^{2}+11\,Aam+28\,Aa \right ) }{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^3+a)*(B*x^3+A),x)

[Out]

x^(1+m)*(B*b*m^2*x^6+5*B*b*m*x^6+4*B*b*x^6+A*b*m^2*x^3+B*a*m^2*x^3+8*A*b*m*x^3+8
*B*a*m*x^3+7*A*b*x^3+7*B*a*x^3+A*a*m^2+11*A*a*m+28*A*a)/(7+m)/(4+m)/(1+m)

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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)*(b*x^3 + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246423, size = 124, normalized size = 2.76 \[ \frac{{\left ({\left (B b m^{2} + 5 \, B b m + 4 \, B b\right )} x^{7} +{\left ({\left (B a + A b\right )} m^{2} + 7 \, B a + 7 \, A b + 8 \,{\left (B a + A b\right )} m\right )} x^{4} +{\left (A a m^{2} + 11 \, A a m + 28 \, A a\right )} x\right )} x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b*m^2 + 5*B*b*m + 4*B*b)*x^7 + ((B*a + A*b)*m^2 + 7*B*a + 7*A*b + 8*(B*a + A
*b)*m)*x^4 + (A*a*m^2 + 11*A*a*m + 28*A*a)*x)*x^m/(m^3 + 12*m^2 + 39*m + 28)

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Sympy [A]  time = 4.45935, size = 410, normalized size = 9.11 \[ \begin{cases} - \frac{A a}{6 x^{6}} - \frac{A b}{3 x^{3}} - \frac{B a}{3 x^{3}} + B b \log{\left (x \right )} & \text{for}\: m = -7 \\- \frac{A a}{3 x^{3}} + A b \log{\left (x \right )} + B a \log{\left (x \right )} + \frac{B b x^{3}}{3} & \text{for}\: m = -4 \\A a \log{\left (x \right )} + \frac{A b x^{3}}{3} + \frac{B a x^{3}}{3} + \frac{B b x^{6}}{6} & \text{for}\: m = -1 \\\frac{A a m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 A a m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 A a x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{A b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{8 A b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{7 A b x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{B a m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{8 B a m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{7 B a x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{B b m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 B b m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 B b x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-A*a/(6*x**6) - A*b/(3*x**3) - B*a/(3*x**3) + B*b*log(x), Eq(m, -7)),
(-A*a/(3*x**3) + A*b*log(x) + B*a*log(x) + B*b*x**3/3, Eq(m, -4)), (A*a*log(x)
+ A*b*x**3/3 + B*a*x**3/3 + B*b*x**6/6, Eq(m, -1)), (A*a*m**2*x*x**m/(m**3 + 12*
m**2 + 39*m + 28) + 11*A*a*m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*A*a*x*x**m
/(m**3 + 12*m**2 + 39*m + 28) + A*b*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28)
+ 8*A*b*m*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 7*A*b*x**4*x**m/(m**3 + 12*m*
*2 + 39*m + 28) + B*a*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 8*B*a*m*x**4
*x**m/(m**3 + 12*m**2 + 39*m + 28) + 7*B*a*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28
) + B*b*m**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 5*B*b*m*x**7*x**m/(m**3 +
12*m**2 + 39*m + 28) + 4*B*b*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28), True))

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GIAC/XCAS [A]  time = 0.216468, size = 225, normalized size = 5. \[ \frac{B b m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, B b m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, B b x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + B a m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + A b m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, B a m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, A b m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 7 \, B a x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 7 \, A b x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + A a m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, A a m x e^{\left (m{\rm ln}\left (x\right )\right )} + 28 \, A a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*m^2*x^7*e^(m*ln(x)) + 5*B*b*m*x^7*e^(m*ln(x)) + 4*B*b*x^7*e^(m*ln(x)) + B*a
*m^2*x^4*e^(m*ln(x)) + A*b*m^2*x^4*e^(m*ln(x)) + 8*B*a*m*x^4*e^(m*ln(x)) + 8*A*b
*m*x^4*e^(m*ln(x)) + 7*B*a*x^4*e^(m*ln(x)) + 7*A*b*x^4*e^(m*ln(x)) + A*a*m^2*x*e
^(m*ln(x)) + 11*A*a*m*x*e^(m*ln(x)) + 28*A*a*x*e^(m*ln(x)))/(m^3 + 12*m^2 + 39*m
+ 28)