3.20.32 \(\int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac {1215}{704} (1-2 x)^{11/2}+\frac {117}{4} (1-2 x)^{9/2}-\frac {13905}{64} (1-2 x)^{7/2}+\frac {7497}{8} (1-2 x)^{5/2}-\frac {173215}{64} (1-2 x)^{3/2}+\frac {60025}{8} \sqrt {1-2 x}+\frac {184877}{64 \sqrt {1-2 x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {1215}{704} (1-2 x)^{11/2}+\frac {117}{4} (1-2 x)^{9/2}-\frac {13905}{64} (1-2 x)^{7/2}+\frac {7497}{8} (1-2 x)^{5/2}-\frac {173215}{64} (1-2 x)^{3/2}+\frac {60025}{8} \sqrt {1-2 x}+\frac {184877}{64 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^5*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

184877/(64*Sqrt[1 - 2*x]) + (60025*Sqrt[1 - 2*x])/8 - (173215*(1 - 2*x)^(3/2))/64 + (7497*(1 - 2*x)^(5/2))/8 -
(13905*(1 - 2*x)^(7/2))/64 + (117*(1 - 2*x)^(9/2))/4 - (1215*(1 - 2*x)^(11/2))/704

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {184877}{64 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {519645}{64} \sqrt {1-2 x}-\frac {37485}{8} (1-2 x)^{3/2}+\frac {97335}{64} (1-2 x)^{5/2}-\frac {1053}{4} (1-2 x)^{7/2}+\frac {1215}{64} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac {184877}{64 \sqrt {1-2 x}}+\frac {60025}{8} \sqrt {1-2 x}-\frac {173215}{64} (1-2 x)^{3/2}+\frac {7497}{8} (1-2 x)^{5/2}-\frac {13905}{64} (1-2 x)^{7/2}+\frac {117}{4} (1-2 x)^{9/2}-\frac {1215}{704} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.47 \begin {gather*} \frac {-1215 x^6-6651 x^5-17055 x^4-28692 x^3-41012 x^2-91704 x+92760}{11 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^5*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

(92760 - 91704*x - 41012*x^2 - 28692*x^3 - 17055*x^4 - 6651*x^5 - 1215*x^6)/(11*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.02, size = 67, normalized size = 0.73 \begin {gather*} \frac {-1215 (1-2 x)^6+20592 (1-2 x)^5-152955 (1-2 x)^4+659736 (1-2 x)^3-1905365 (1-2 x)^2+5282200 (1-2 x)+2033647}{704 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + 3*x)^5*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

(2033647 + 5282200*(1 - 2*x) - 1905365*(1 - 2*x)^2 + 659736*(1 - 2*x)^3 - 152955*(1 - 2*x)^4 + 20592*(1 - 2*x)
^5 - 1215*(1 - 2*x)^6)/(704*Sqrt[1 - 2*x])

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fricas [A]  time = 1.54, size = 46, normalized size = 0.50 \begin {gather*} \frac {{\left (1215 \, x^{6} + 6651 \, x^{5} + 17055 \, x^{4} + 28692 \, x^{3} + 41012 \, x^{2} + 91704 \, x - 92760\right )} \sqrt {-2 \, x + 1}}{11 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/11*(1215*x^6 + 6651*x^5 + 17055*x^4 + 28692*x^3 + 41012*x^2 + 91704*x - 92760)*sqrt(-2*x + 1)/(2*x - 1)

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giac [A]  time = 1.24, size = 92, normalized size = 1.00 \begin {gather*} \frac {1215}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {117}{4} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {13905}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {7497}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {173215}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {60025}{8} \, \sqrt {-2 \, x + 1} + \frac {184877}{64 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

1215/704*(2*x - 1)^5*sqrt(-2*x + 1) + 117/4*(2*x - 1)^4*sqrt(-2*x + 1) + 13905/64*(2*x - 1)^3*sqrt(-2*x + 1) +
7497/8*(2*x - 1)^2*sqrt(-2*x + 1) - 173215/64*(-2*x + 1)^(3/2) + 60025/8*sqrt(-2*x + 1) + 184877/64/sqrt(-2*x
+ 1)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {1215 x^{6}+6651 x^{5}+17055 x^{4}+28692 x^{3}+41012 x^{2}+91704 x -92760}{11 \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^5*(5*x+3)/(-2*x+1)^(3/2),x)

[Out]

-1/11*(1215*x^6+6651*x^5+17055*x^4+28692*x^3+41012*x^2+91704*x-92760)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.45, size = 64, normalized size = 0.70 \begin {gather*} -\frac {1215}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {117}{4} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {13905}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {7497}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {173215}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {60025}{8} \, \sqrt {-2 \, x + 1} + \frac {184877}{64 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-1215/704*(-2*x + 1)^(11/2) + 117/4*(-2*x + 1)^(9/2) - 13905/64*(-2*x + 1)^(7/2) + 7497/8*(-2*x + 1)^(5/2) - 1
73215/64*(-2*x + 1)^(3/2) + 60025/8*sqrt(-2*x + 1) + 184877/64/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {184877}{64\,\sqrt {1-2\,x}}+\frac {60025\,\sqrt {1-2\,x}}{8}-\frac {173215\,{\left (1-2\,x\right )}^{3/2}}{64}+\frac {7497\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {13905\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {117\,{\left (1-2\,x\right )}^{9/2}}{4}-\frac {1215\,{\left (1-2\,x\right )}^{11/2}}{704} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^5*(5*x + 3))/(1 - 2*x)^(3/2),x)

[Out]

184877/(64*(1 - 2*x)^(1/2)) + (60025*(1 - 2*x)^(1/2))/8 - (173215*(1 - 2*x)^(3/2))/64 + (7497*(1 - 2*x)^(5/2))
/8 - (13905*(1 - 2*x)^(7/2))/64 + (117*(1 - 2*x)^(9/2))/4 - (1215*(1 - 2*x)^(11/2))/704

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sympy [A]  time = 42.54, size = 82, normalized size = 0.89 \begin {gather*} - \frac {1215 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {117 \left (1 - 2 x\right )^{\frac {9}{2}}}{4} - \frac {13905 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {7497 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {173215 \left (1 - 2 x\right )^{\frac {3}{2}}}{64} + \frac {60025 \sqrt {1 - 2 x}}{8} + \frac {184877}{64 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**5*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

-1215*(1 - 2*x)**(11/2)/704 + 117*(1 - 2*x)**(9/2)/4 - 13905*(1 - 2*x)**(7/2)/64 + 7497*(1 - 2*x)**(5/2)/8 - 1
73215*(1 - 2*x)**(3/2)/64 + 60025*sqrt(1 - 2*x)/8 + 184877/(64*sqrt(1 - 2*x))

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