3.15.38 \(\int x^m (a+b x^7)^2 \, dx\) [1438]

Optimal. Leaf size=43 \[ \frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{8+m}}{8+m}+\frac {b^2 x^{15+m}}{15+m} \]

[Out]

a^2*x^(1+m)/(1+m)+2*a*b*x^(8+m)/(8+m)+b^2*x^(15+m)/(15+m)

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} \frac {a^2 x^{m+1}}{m+1}+\frac {2 a b x^{m+8}}{m+8}+\frac {b^2 x^{m+15}}{m+15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^7)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(8 + m))/(8 + m) + (b^2*x^(15 + m))/(15 + m)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^m \left (a+b x^7\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{7+m}+b^2 x^{14+m}\right ) \, dx\\ &=\frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{8+m}}{8+m}+\frac {b^2 x^{15+m}}{15+m}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 40, normalized size = 0.93 \begin {gather*} x^{1+m} \left (\frac {a^2}{1+m}+\frac {2 a b x^7}{8+m}+\frac {b^2 x^{14}}{15+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^7)^2,x]

[Out]

x^(1 + m)*(a^2/(1 + m) + (2*a*b*x^7)/(8 + m) + (b^2*x^14)/(15 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(43)=86\).
time = 0.16, size = 92, normalized size = 2.14

method	result	size
risch	\(\frac {x \left (b^{2} m^{2} x^{14}+9 m \,x^{14} b^{2}+8 b^{2} x^{14}+2 a b \,m^{2} x^{7}+32 m \,x^{7} a b +30 a b \,x^{7}+a^{2} m^{2}+23 m \,a^{2}+120 a^{2}\right ) x^{m}}{\left (1+m \right ) \left (8+m \right ) \left (15+m \right )}\)	\(92\)
gosper	\(\frac {x^{1+m} \left (b^{2} m^{2} x^{14}+9 m \,x^{14} b^{2}+8 b^{2} x^{14}+2 a b \,m^{2} x^{7}+32 m \,x^{7} a b +30 a b \,x^{7}+a^{2} m^{2}+23 m \,a^{2}+120 a^{2}\right )}{\left (1+m \right ) \left (8+m \right ) \left (15+m \right )}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^7+a)^2,x,method=_RETURNVERBOSE)

[Out]

x*(b^2*m^2*x^14+9*b^2*m*x^14+8*b^2*x^14+2*a*b*m^2*x^7+32*a*b*m*x^7+30*a*b*x^7+a^2*m^2+23*a^2*m+120*a^2)*x^m/(1
+m)/(8+m)/(15+m)

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Maxima [A]
time = 0.29, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{m + 15}}{m + 15} + \frac {2 \, a b x^{m + 8}}{m + 8} + \frac {a^{2} x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^7+a)^2,x, algorithm="maxima")

[Out]

b^2*x^(m + 15)/(m + 15) + 2*a*b*x^(m + 8)/(m + 8) + a^2*x^(m + 1)/(m + 1)

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Fricas [A]
time = 0.37, size = 85, normalized size = 1.98 \begin {gather*} \frac {{\left ({\left (b^{2} m^{2} + 9 \, b^{2} m + 8 \, b^{2}\right )} x^{15} + 2 \, {\left (a b m^{2} + 16 \, a b m + 15 \, a b\right )} x^{8} + {\left (a^{2} m^{2} + 23 \, a^{2} m + 120 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^7+a)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 9*b^2*m + 8*b^2)*x^15 + 2*(a*b*m^2 + 16*a*b*m + 15*a*b)*x^8 + (a^2*m^2 + 23*a^2*m + 120*a^2)*x)*x^
m/(m^3 + 24*m^2 + 143*m + 120)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (36) = 72\).
time = 0.88, size = 313, normalized size = 7.28 \begin {gather*} \begin {cases} - \frac {a^{2}}{14 x^{14}} - \frac {2 a b}{7 x^{7}} + b^{2} \log {\left (x \right )} & \text {for}\: m = -15 \\- \frac {a^{2}}{7 x^{7}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{7}}{7} & \text {for}\: m = -8 \\a^{2} \log {\left (x \right )} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{14}}{14} & \text {for}\: m = -1 \\\frac {a^{2} m^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {23 a^{2} m x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {120 a^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {2 a b m^{2} x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {32 a b m x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {30 a b x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {b^{2} m^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {9 b^{2} m x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {8 b^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**7+a)**2,x)

[Out]

Piecewise((-a**2/(14*x**14) - 2*a*b/(7*x**7) + b**2*log(x), Eq(m, -15)), (-a**2/(7*x**7) + 2*a*b*log(x) + b**2
*x**7/7, Eq(m, -8)), (a**2*log(x) + 2*a*b*x**7/7 + b**2*x**14/14, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 24*m**
2 + 143*m + 120) + 23*a**2*m*x*x**m/(m**3 + 24*m**2 + 143*m + 120) + 120*a**2*x*x**m/(m**3 + 24*m**2 + 143*m +
120) + 2*a*b*m**2*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + 32*a*b*m*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120
) + 30*a*b*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + b**2*m**2*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 9*
b**2*m*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 8*b**2*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (43) = 86\).
time = 1.54, size = 117, normalized size = 2.72 \begin {gather*} \frac {b^{2} m^{2} x^{15} x^{m} + 9 \, b^{2} m x^{15} x^{m} + 8 \, b^{2} x^{15} x^{m} + 2 \, a b m^{2} x^{8} x^{m} + 32 \, a b m x^{8} x^{m} + 30 \, a b x^{8} x^{m} + a^{2} m^{2} x x^{m} + 23 \, a^{2} m x x^{m} + 120 \, a^{2} x x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^7+a)^2,x, algorithm="giac")

[Out]

(b^2*m^2*x^15*x^m + 9*b^2*m*x^15*x^m + 8*b^2*x^15*x^m + 2*a*b*m^2*x^8*x^m + 32*a*b*m*x^8*x^m + 30*a*b*x^8*x^m
+ a^2*m^2*x*x^m + 23*a^2*m*x*x^m + 120*a^2*x*x^m)/(m^3 + 24*m^2 + 143*m + 120)

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Mupad [B]
time = 1.37, size = 93, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {a^2\,x\,\left (m^2+23\,m+120\right )}{m^3+24\,m^2+143\,m+120}+\frac {b^2\,x^{15}\,\left (m^2+9\,m+8\right )}{m^3+24\,m^2+143\,m+120}+\frac {2\,a\,b\,x^8\,\left (m^2+16\,m+15\right )}{m^3+24\,m^2+143\,m+120}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^7)^2,x)

[Out]

x^m*((a^2*x*(23*m + m^2 + 120))/(143*m + 24*m^2 + m^3 + 120) + (b^2*x^15*(9*m + m^2 + 8))/(143*m + 24*m^2 + m^
3 + 120) + (2*a*b*x^8*(16*m + m^2 + 15))/(143*m + 24*m^2 + m^3 + 120))

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