3.18.91 \(\int (1-2 x)^{5/2} (2+3 x)^4 (3+5 x) \, dx\)

Optimal. Leaf size=79 \[ \frac {405}{544} (1-2 x)^{17/2}-\frac {1557}{160} (1-2 x)^{15/2}+\frac {10773}{208} (1-2 x)^{13/2}-\frac {24843}{176} (1-2 x)^{11/2}+\frac {57281}{288} (1-2 x)^{9/2}-\frac {3773}{32} (1-2 x)^{7/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 79, 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 {405}{544} (1-2 x)^{17/2}-\frac {1557}{160} (1-2 x)^{15/2}+\frac {10773}{208} (1-2 x)^{13/2}-\frac {24843}{176} (1-2 x)^{11/2}+\frac {57281}{288} (1-2 x)^{9/2}-\frac {3773}{32} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-3773*(1 - 2*x)^(7/2))/32 + (57281*(1 - 2*x)^(9/2))/288 - (24843*(1 - 2*x)^(11/2))/176 + (10773*(1 - 2*x)^(13
/2))/208 - (1557*(1 - 2*x)^(15/2))/160 + (405*(1 - 2*x)^(17/2))/544

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 (1-2 x)^{5/2} (2+3 x)^4 (3+5 x) \, dx &=\int \left (\frac {26411}{32} (1-2 x)^{5/2}-\frac {57281}{32} (1-2 x)^{7/2}+\frac {24843}{16} (1-2 x)^{9/2}-\frac {10773}{16} (1-2 x)^{11/2}+\frac {4671}{32} (1-2 x)^{13/2}-\frac {405}{32} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {3773}{32} (1-2 x)^{7/2}+\frac {57281}{288} (1-2 x)^{9/2}-\frac {24843}{176} (1-2 x)^{11/2}+\frac {10773}{208} (1-2 x)^{13/2}-\frac {1557}{160} (1-2 x)^{15/2}+\frac {405}{544} (1-2 x)^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{7/2} \left (2606175 x^5+10517364 x^4+17777232 x^3+16066296 x^2+8043328 x+1899184\right )}{109395} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

-1/109395*((1 - 2*x)^(7/2)*(1899184 + 8043328*x + 16066296*x^2 + 17777232*x^3 + 10517364*x^4 + 2606175*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {2606175 (1-2 x)^{17/2}-34065603 (1-2 x)^{15/2}[phone] (1-2 x)^{13/2}-494127270 (1-2 x)^{11/2}[phone] (1-2 x)^{9/2}-412747335 (1-2 x)^{7/2}}{3500640} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)^(5/2)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-412747335*(1 - 2*x)^(7/2) + 696250555*(1 - 2*x)^(9/2) - 494127270*(1 - 2*x)^(11/2) + 181309590*(1 - 2*x)^(13
/2) - 34065603*(1 - 2*x)^(15/2) + 2606175*(1 - 2*x)^(17/2))/3500640

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fricas [A]  time = 1.28, size = 49, normalized size = 0.62 \begin {gather*} \frac {1}{109395} \, {\left (20849400 \, x^{8} + 52864812 \, x^{7} + 31646538 \, x^{6} - 24298407 \, x^{5} - 32302900 \, x^{4} - 2705920 \, x^{3} + 9403464 \, x^{2} + 3351776 \, x - 1899184\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x),x, algorithm="fricas")

[Out]

1/109395*(20849400*x^8 + 52864812*x^7 + 31646538*x^6 - 24298407*x^5 - 32302900*x^4 - 2705920*x^3 + 9403464*x^2
+ 3351776*x - 1899184)*sqrt(-2*x + 1)

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giac [A]  time = 0.99, size = 97, normalized size = 1.23 \begin {gather*} \frac {405}{544} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {1557}{160} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {10773}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {24843}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {57281}{288} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3773}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x),x, algorithm="giac")

[Out]

405/544*(2*x - 1)^8*sqrt(-2*x + 1) + 1557/160*(2*x - 1)^7*sqrt(-2*x + 1) + 10773/208*(2*x - 1)^6*sqrt(-2*x + 1
) + 24843/176*(2*x - 1)^5*sqrt(-2*x + 1) + 57281/288*(2*x - 1)^4*sqrt(-2*x + 1) + 3773/32*(2*x - 1)^3*sqrt(-2*
x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (2606175 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}+8043328 x +1899184\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{109395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(3*x+2)^4*(5*x+3),x)

[Out]

-1/109395*(2606175*x^5+10517364*x^4+17777232*x^3+16066296*x^2+8043328*x+1899184)*(-2*x+1)^(7/2)

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maxima [A]  time = 0.49, size = 55, normalized size = 0.70 \begin {gather*} \frac {405}{544} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {1557}{160} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {10773}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {24843}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {57281}{288} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3773}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x),x, algorithm="maxima")

[Out]

405/544*(-2*x + 1)^(17/2) - 1557/160*(-2*x + 1)^(15/2) + 10773/208*(-2*x + 1)^(13/2) - 24843/176*(-2*x + 1)^(1
1/2) + 57281/288*(-2*x + 1)^(9/2) - 3773/32*(-2*x + 1)^(7/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {57281\,{\left (1-2\,x\right )}^{9/2}}{288}-\frac {3773\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {24843\,{\left (1-2\,x\right )}^{11/2}}{176}+\frac {10773\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {1557\,{\left (1-2\,x\right )}^{15/2}}{160}+\frac {405\,{\left (1-2\,x\right )}^{17/2}}{544} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)*(3*x + 2)^4*(5*x + 3),x)

[Out]

(57281*(1 - 2*x)^(9/2))/288 - (3773*(1 - 2*x)^(7/2))/32 - (24843*(1 - 2*x)^(11/2))/176 + (10773*(1 - 2*x)^(13/
2))/208 - (1557*(1 - 2*x)^(15/2))/160 + (405*(1 - 2*x)^(17/2))/544

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sympy [A]  time = 21.44, size = 70, normalized size = 0.89 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {17}{2}}}{544} - \frac {1557 \left (1 - 2 x\right )^{\frac {15}{2}}}{160} + \frac {10773 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {24843 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {57281 \left (1 - 2 x\right )^{\frac {9}{2}}}{288} - \frac {3773 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(2+3*x)**4*(3+5*x),x)

[Out]

405*(1 - 2*x)**(17/2)/544 - 1557*(1 - 2*x)**(15/2)/160 + 10773*(1 - 2*x)**(13/2)/208 - 24843*(1 - 2*x)**(11/2)
/176 + 57281*(1 - 2*x)**(9/2)/288 - 3773*(1 - 2*x)**(7/2)/32

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