3.97.16 \(\int (-6750000+e^x+10845000 x-7095600 x^2+2436696 x^3-473040 x^4+52056 x^5-3024 x^6+72 x^7) \, dx\)

Optimal. Leaf size=18 \[ 3+e^x+9 \left ((-5+x)^2-2 x\right )^4 \]

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Rubi [B]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 2.33, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2194} \begin {gather*} 9 x^8-432 x^7+8676 x^6-94608 x^5+609174 x^4-2365200 x^3+5422500 x^2-6750000 x+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-6750000 + E^x + 10845000*x - 7095600*x^2 + 2436696*x^3 - 473040*x^4 + 52056*x^5 - 3024*x^6 + 72*x^7,x]

[Out]

E^x - 6750000*x + 5422500*x^2 - 2365200*x^3 + 609174*x^4 - 94608*x^5 + 8676*x^6 - 432*x^7 + 9*x^8

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-6750000 x+5422500 x^2-2365200 x^3+609174 x^4-94608 x^5+8676 x^6-432 x^7+9 x^8+\int e^x \, dx\\ &=e^x-6750000 x+5422500 x^2-2365200 x^3+609174 x^4-94608 x^5+8676 x^6-432 x^7+9 x^8\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 42, normalized size = 2.33 \begin {gather*} e^x-6750000 x+5422500 x^2-2365200 x^3+609174 x^4-94608 x^5+8676 x^6-432 x^7+9 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-6750000 + E^x + 10845000*x - 7095600*x^2 + 2436696*x^3 - 473040*x^4 + 52056*x^5 - 3024*x^6 + 72*x^7
,x]

[Out]

E^x - 6750000*x + 5422500*x^2 - 2365200*x^3 + 609174*x^4 - 94608*x^5 + 8676*x^6 - 432*x^7 + 9*x^8

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fricas [B]  time = 0.54, size = 41, normalized size = 2.28 \begin {gather*} 9 \, x^{8} - 432 \, x^{7} + 8676 \, x^{6} - 94608 \, x^{5} + 609174 \, x^{4} - 2365200 \, x^{3} + 5422500 \, x^{2} - 6750000 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+72*x^7-3024*x^6+52056*x^5-473040*x^4+2436696*x^3-7095600*x^2+10845000*x-6750000,x, algorithm=
"fricas")

[Out]

9*x^8 - 432*x^7 + 8676*x^6 - 94608*x^5 + 609174*x^4 - 2365200*x^3 + 5422500*x^2 - 6750000*x + e^x

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giac [B]  time = 0.14, size = 41, normalized size = 2.28 \begin {gather*} 9 \, x^{8} - 432 \, x^{7} + 8676 \, x^{6} - 94608 \, x^{5} + 609174 \, x^{4} - 2365200 \, x^{3} + 5422500 \, x^{2} - 6750000 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+72*x^7-3024*x^6+52056*x^5-473040*x^4+2436696*x^3-7095600*x^2+10845000*x-6750000,x, algorithm=
"giac")

[Out]

9*x^8 - 432*x^7 + 8676*x^6 - 94608*x^5 + 609174*x^4 - 2365200*x^3 + 5422500*x^2 - 6750000*x + e^x

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maple [B]  time = 0.02, size = 42, normalized size = 2.33

method	result	size
default	\(9 x^{8}-432 x^{7}+8676 x^{6}-94608 x^{5}+609174 x^{4}-2365200 x^{3}+5422500 x^{2}-6750000 x +{\mathrm e}^{x}\)	\(42\)
norman	\(9 x^{8}-432 x^{7}+8676 x^{6}-94608 x^{5}+609174 x^{4}-2365200 x^{3}+5422500 x^{2}-6750000 x +{\mathrm e}^{x}\)	\(42\)
risch	\(9 x^{8}-432 x^{7}+8676 x^{6}-94608 x^{5}+609174 x^{4}-2365200 x^{3}+5422500 x^{2}-6750000 x +{\mathrm e}^{x}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)+72*x^7-3024*x^6+52056*x^5-473040*x^4+2436696*x^3-7095600*x^2+10845000*x-6750000,x,method=_RETURNVER
BOSE)

[Out]

9*x^8-432*x^7+8676*x^6-94608*x^5+609174*x^4-2365200*x^3+5422500*x^2-6750000*x+exp(x)

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maxima [B]  time = 0.37, size = 41, normalized size = 2.28 \begin {gather*} 9 \, x^{8} - 432 \, x^{7} + 8676 \, x^{6} - 94608 \, x^{5} + 609174 \, x^{4} - 2365200 \, x^{3} + 5422500 \, x^{2} - 6750000 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+72*x^7-3024*x^6+52056*x^5-473040*x^4+2436696*x^3-7095600*x^2+10845000*x-6750000,x, algorithm=
"maxima")

[Out]

9*x^8 - 432*x^7 + 8676*x^6 - 94608*x^5 + 609174*x^4 - 2365200*x^3 + 5422500*x^2 - 6750000*x + e^x

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mupad [B]  time = 5.85, size = 41, normalized size = 2.28 \begin {gather*} {\mathrm {e}}^x-6750000\,x+5422500\,x^2-2365200\,x^3+609174\,x^4-94608\,x^5+8676\,x^6-432\,x^7+9\,x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10845000*x + exp(x) - 7095600*x^2 + 2436696*x^3 - 473040*x^4 + 52056*x^5 - 3024*x^6 + 72*x^7 - 6750000,x)

[Out]

exp(x) - 6750000*x + 5422500*x^2 - 2365200*x^3 + 609174*x^4 - 94608*x^5 + 8676*x^6 - 432*x^7 + 9*x^8

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sympy [B]  time = 0.09, size = 41, normalized size = 2.28 \begin {gather*} 9 x^{8} - 432 x^{7} + 8676 x^{6} - 94608 x^{5} + 609174 x^{4} - 2365200 x^{3} + 5422500 x^{2} - 6750000 x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+72*x**7-3024*x**6+52056*x**5-473040*x**4+2436696*x**3-7095600*x**2+10845000*x-6750000,x)

[Out]

9*x**8 - 432*x**7 + 8676*x**6 - 94608*x**5 + 609174*x**4 - 2365200*x**3 + 5422500*x**2 - 6750000*x + exp(x)

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