3.3.5 \(\int x^{15} (c+d x)^7 (2 c+3 d x) \, dx\) [205]

Optimal. Leaf size=14 \[ \frac {1}{8} x^{16} (c+d x)^8 \]

[Out]

1/8*x^16*(d*x+c)^8

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {75} \begin {gather*} \frac {1}{8} x^{16} (c+d x)^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^15*(c + d*x)^7*(2*c + 3*d*x),x]

[Out]

(x^16*(c + d*x)^8)/8

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin {align*} \int x^{15} (c+d x)^7 (2 c+3 d x) \, dx &=\frac {1}{8} x^{16} (c+d x)^8\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(14)=28\).
time = 0.00, size = 98, normalized size = 7.00 \begin {gather*} \frac {c^8 x^{16}}{8}+c^7 d x^{17}+\frac {7}{2} c^6 d^2 x^{18}+7 c^5 d^3 x^{19}+\frac {35}{4} c^4 d^4 x^{20}+7 c^3 d^5 x^{21}+\frac {7}{2} c^2 d^6 x^{22}+c d^7 x^{23}+\frac {d^8 x^{24}}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^15*(c + d*x)^7*(2*c + 3*d*x),x]

[Out]

(c^8*x^16)/8 + c^7*d*x^17 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20)/4 + 7*c^3*d^5*x^21 + (7*c^
2*d^6*x^22)/2 + c*d^7*x^23 + (d^8*x^24)/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(12)=24\).
time = 0.19, size = 89, normalized size = 6.36

method	result	size
gosper	\(\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}+\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}\)	\(89\)
default	\(\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}+\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}\)	\(89\)
norman	\(\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}+\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}\)	\(89\)
risch	\(\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}+\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(d*x+c)^7*(3*d*x+2*c),x,method=_RETURNVERBOSE)

[Out]

1/8*x^16*c^8+c^7*d*x^17+7/2*x^18*c^6*d^2+7*c^5*d^3*x^19+35/4*x^20*c^4*d^4+7*c^3*d^5*x^21+7/2*x^22*c^2*d^6+c*d^
7*x^23+1/8*d^8*x^24

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).
time = 0.27, size = 88, normalized size = 6.29 \begin {gather*} \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(d*x+c)^7*(3*d*x+2*c),x, algorithm="maxima")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).
time = 0.39, size = 88, normalized size = 6.29 \begin {gather*} \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(d*x+c)^7*(3*d*x+2*c),x, algorithm="fricas")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (10) = 20\).
time = 0.02, size = 97, normalized size = 6.93 \begin {gather*} \frac {c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac {7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac {35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac {7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15*(d*x+c)**7*(3*d*x+2*c),x)

[Out]

c**8*x**16/8 + c**7*d*x**17 + 7*c**6*d**2*x**18/2 + 7*c**5*d**3*x**19 + 35*c**4*d**4*x**20/4 + 7*c**3*d**5*x**
21 + 7*c**2*d**6*x**22/2 + c*d**7*x**23 + d**8*x**24/8

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).
time = 4.19, size = 88, normalized size = 6.29 \begin {gather*} \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(d*x+c)^7*(3*d*x+2*c),x, algorithm="giac")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

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Mupad [B]
time = 0.04, size = 88, normalized size = 6.29 \begin {gather*} \frac {c^8\,x^{16}}{8}+c^7\,d\,x^{17}+\frac {7\,c^6\,d^2\,x^{18}}{2}+7\,c^5\,d^3\,x^{19}+\frac {35\,c^4\,d^4\,x^{20}}{4}+7\,c^3\,d^5\,x^{21}+\frac {7\,c^2\,d^6\,x^{22}}{2}+c\,d^7\,x^{23}+\frac {d^8\,x^{24}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(2*c + 3*d*x)*(c + d*x)^7,x)

[Out]

(c^8*x^16)/8 + (d^8*x^24)/8 + c^7*d*x^17 + c*d^7*x^23 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20
)/4 + 7*c^3*d^5*x^21 + (7*c^2*d^6*x^22)/2

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