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

Optimal. Leaf size=66 \[ -\frac {225}{208} (1-2 x)^{13/2}+\frac {255}{22} (1-2 x)^{11/2}-\frac {3467}{72} (1-2 x)^{9/2}+\frac {187}{2} (1-2 x)^{7/2}-\frac {5929}{80} (1-2 x)^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {88} \begin {gather*} -\frac {225}{208} (1-2 x)^{13/2}+\frac {255}{22} (1-2 x)^{11/2}-\frac {3467}{72} (1-2 x)^{9/2}+\frac {187}{2} (1-2 x)^{7/2}-\frac {5929}{80} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*(1 - 2*x)^(5/2))/80 + (187*(1 - 2*x)^(7/2))/2 - (3467*(1 - 2*x)^(9/2))/72 + (255*(1 - 2*x)^(11/2))/22 -
(225*(1 - 2*x)^(13/2))/208

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2 \, dx &=\int \left (\frac {5929}{16} (1-2 x)^{3/2}-\frac {1309}{2} (1-2 x)^{5/2}+\frac {3467}{8} (1-2 x)^{7/2}-\frac {255}{2} (1-2 x)^{9/2}+\frac {225}{16} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {5929}{80} (1-2 x)^{5/2}+\frac {187}{2} (1-2 x)^{7/2}-\frac {3467}{72} (1-2 x)^{9/2}+\frac {255}{22} (1-2 x)^{11/2}-\frac {225}{208} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 33, normalized size = 0.50 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (111375 x^4+373950 x^3+511465 x^2+355730 x+117478\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6435*((1 - 2*x)^(5/2)*(117478 + 355730*x + 511465*x^2 + 373950*x^3 + 111375*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {-111375 (1-2 x)^{13/2}+1193400 (1-2 x)^{11/2}-4957810 (1-2 x)^{9/2}+9626760 (1-2 x)^{7/2}-7630623 (1-2 x)^{5/2}}{102960} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7630623*(1 - 2*x)^(5/2) + 9626760*(1 - 2*x)^(7/2) - 4957810*(1 - 2*x)^(9/2) + 1193400*(1 - 2*x)^(11/2) - 111
375*(1 - 2*x)^(13/2))/102960

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fricas [A]  time = 1.44, size = 39, normalized size = 0.59 \begin {gather*} -\frac {1}{6435} \, {\left (445500 \, x^{6} + 1050300 \, x^{5} + 661435 \, x^{4} - 248990 \, x^{3} - 441543 \, x^{2} - 114182 \, x + 117478\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6435*(445500*x^6 + 1050300*x^5 + 661435*x^4 - 248990*x^3 - 441543*x^2 - 114182*x + 117478)*sqrt(-2*x + 1)

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giac [A]  time = 0.92, size = 81, normalized size = 1.23 \begin {gather*} -\frac {225}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {255}{22} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {3467}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {187}{2} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {5929}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225/208*(2*x - 1)^6*sqrt(-2*x + 1) - 255/22*(2*x - 1)^5*sqrt(-2*x + 1) - 3467/72*(2*x - 1)^4*sqrt(-2*x + 1) -
187/2*(2*x - 1)^3*sqrt(-2*x + 1) - 5929/80*(2*x - 1)^2*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (111375 x^{4}+373950 x^{3}+511465 x^{2}+355730 x +117478\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{6435} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6435*(111375*x^4+373950*x^3+511465*x^2+355730*x+117478)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.55, size = 46, normalized size = 0.70 \begin {gather*} -\frac {225}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {255}{22} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {3467}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {187}{2} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {5929}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225/208*(-2*x + 1)^(13/2) + 255/22*(-2*x + 1)^(11/2) - 3467/72*(-2*x + 1)^(9/2) + 187/2*(-2*x + 1)^(7/2) - 59
29/80*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {187\,{\left (1-2\,x\right )}^{7/2}}{2}-\frac {5929\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {3467\,{\left (1-2\,x\right )}^{9/2}}{72}+\frac {255\,{\left (1-2\,x\right )}^{11/2}}{22}-\frac {225\,{\left (1-2\,x\right )}^{13/2}}{208} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(187*(1 - 2*x)^(7/2))/2 - (5929*(1 - 2*x)^(5/2))/80 - (3467*(1 - 2*x)^(9/2))/72 + (255*(1 - 2*x)^(11/2))/22 -
(225*(1 - 2*x)^(13/2))/208

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sympy [A]  time = 13.49, size = 58, normalized size = 0.88 \begin {gather*} - \frac {225 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} + \frac {255 \left (1 - 2 x\right )^{\frac {11}{2}}}{22} - \frac {3467 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} + \frac {187 \left (1 - 2 x\right )^{\frac {7}{2}}}{2} - \frac {5929 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225*(1 - 2*x)**(13/2)/208 + 255*(1 - 2*x)**(11/2)/22 - 3467*(1 - 2*x)**(9/2)/72 + 187*(1 - 2*x)**(7/2)/2 - 59
29*(1 - 2*x)**(5/2)/80

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