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

Optimal. Leaf size=79 \[ \frac {75}{32} (1-2 x)^{15/2}-\frac {975}{32} (1-2 x)^{13/2}+\frac {28555}{176} (1-2 x)^{11/2}-\frac {21439}{48} (1-2 x)^{9/2}+\frac {20691}{32} (1-2 x)^{7/2}-\frac {65219}{160} (1-2 x)^{5/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 79, 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 {75}{32} (1-2 x)^{15/2}-\frac {975}{32} (1-2 x)^{13/2}+\frac {28555}{176} (1-2 x)^{11/2}-\frac {21439}{48} (1-2 x)^{9/2}+\frac {20691}{32} (1-2 x)^{7/2}-\frac {65219}{160} (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)^3,x]

[Out]

(-65219*(1 - 2*x)^(5/2))/160 + (20691*(1 - 2*x)^(7/2))/32 - (21439*(1 - 2*x)^(9/2))/48 + (28555*(1 - 2*x)^(11/
2))/176 - (975*(1 - 2*x)^(13/2))/32 + (75*(1 - 2*x)^(15/2))/32

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)^3 \, dx &=\int \left (\frac {65219}{32} (1-2 x)^{3/2}-\frac {144837}{32} (1-2 x)^{5/2}+\frac {64317}{16} (1-2 x)^{7/2}-\frac {28555}{16} (1-2 x)^{9/2}+\frac {12675}{32} (1-2 x)^{11/2}-\frac {1125}{32} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {65219}{160} (1-2 x)^{5/2}+\frac {20691}{32} (1-2 x)^{7/2}-\frac {21439}{48} (1-2 x)^{9/2}+\frac {28555}{176} (1-2 x)^{11/2}-\frac {975}{32} (1-2 x)^{13/2}+\frac {75}{32} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {1}{165} (1-2 x)^{5/2} \left (12375 x^5+49500 x^4+84225 x^3+78730 x^2+42860 x+12136\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/165*((1 - 2*x)^(5/2)*(12136 + 42860*x + 78730*x^2 + 84225*x^3 + 49500*x^4 + 12375*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 58, normalized size = 0.73 \begin {gather*} \frac {\left (12375 (1-2 x)^5-160875 (1-2 x)^4+856650 (1-2 x)^3-2358290 (1-2 x)^2+3414015 (1-2 x)-2152227\right ) (1-2 x)^{5/2}}{5280} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2152227 + 3414015*(1 - 2*x) - 2358290*(1 - 2*x)^2 + 856650*(1 - 2*x)^3 - 160875*(1 - 2*x)^4 + 12375*(1 - 2*
x)^5)*(1 - 2*x)^(5/2))/5280

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fricas [A]  time = 1.38, size = 44, normalized size = 0.56 \begin {gather*} -\frac {1}{165} \, {\left (49500 \, x^{7} + 148500 \, x^{6} + 151275 \, x^{5} + 27520 \, x^{4} - 59255 \, x^{3} - 44166 \, x^{2} - 5684 \, x + 12136\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)^3,x, algorithm="fricas")

[Out]

-1/165*(49500*x^7 + 148500*x^6 + 151275*x^5 + 27520*x^4 - 59255*x^3 - 44166*x^2 - 5684*x + 12136)*sqrt(-2*x +
1)

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giac [A]  time = 0.92, size = 97, normalized size = 1.23 \begin {gather*} -\frac {75}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {975}{32} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {28555}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {21439}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {20691}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {65219}{160} \, {\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)^3,x, algorithm="giac")

[Out]

-75/32*(2*x - 1)^7*sqrt(-2*x + 1) - 975/32*(2*x - 1)^6*sqrt(-2*x + 1) - 28555/176*(2*x - 1)^5*sqrt(-2*x + 1) -
21439/48*(2*x - 1)^4*sqrt(-2*x + 1) - 20691/32*(2*x - 1)^3*sqrt(-2*x + 1) - 65219/160*(2*x - 1)^2*sqrt(-2*x +
1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (12375 x^{5}+49500 x^{4}+84225 x^{3}+78730 x^{2}+42860 x +12136\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{165} \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)^3,x)

[Out]

-1/165*(12375*x^5+49500*x^4+84225*x^3+78730*x^2+42860*x+12136)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.72, size = 55, normalized size = 0.70 \begin {gather*} \frac {75}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {975}{32} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {28555}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {21439}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {20691}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {65219}{160} \, {\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)^3,x, algorithm="maxima")

[Out]

75/32*(-2*x + 1)^(15/2) - 975/32*(-2*x + 1)^(13/2) + 28555/176*(-2*x + 1)^(11/2) - 21439/48*(-2*x + 1)^(9/2) +
20691/32*(-2*x + 1)^(7/2) - 65219/160*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {20691\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {65219\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {21439\,{\left (1-2\,x\right )}^{9/2}}{48}+\frac {28555\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {975\,{\left (1-2\,x\right )}^{13/2}}{32}+\frac {75\,{\left (1-2\,x\right )}^{15/2}}{32} \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)^3,x)

[Out]

(20691*(1 - 2*x)^(7/2))/32 - (65219*(1 - 2*x)^(5/2))/160 - (21439*(1 - 2*x)^(9/2))/48 + (28555*(1 - 2*x)^(11/2
))/176 - (975*(1 - 2*x)^(13/2))/32 + (75*(1 - 2*x)^(15/2))/32

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sympy [A]  time = 16.84, size = 70, normalized size = 0.89 \begin {gather*} \frac {75 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {975 \left (1 - 2 x\right )^{\frac {13}{2}}}{32} + \frac {28555 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {21439 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {20691 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {65219 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} \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)**3,x)

[Out]

75*(1 - 2*x)**(15/2)/32 - 975*(1 - 2*x)**(13/2)/32 + 28555*(1 - 2*x)**(11/2)/176 - 21439*(1 - 2*x)**(9/2)/48 +
20691*(1 - 2*x)**(7/2)/32 - 65219*(1 - 2*x)**(5/2)/160

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