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

Optimal. Leaf size=79 \[ \frac {45}{32} (1-2 x)^{15/2}-\frac {7695}{416} (1-2 x)^{13/2}+\frac {17541}{176} (1-2 x)^{11/2}-\frac {39977}{144} (1-2 x)^{9/2}+\frac {13013}{32} (1-2 x)^{7/2}-\frac {41503}{160} (1-2 x)^{5/2} \]

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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {88} \begin {gather*} \frac {45}{32} (1-2 x)^{15/2}-\frac {7695}{416} (1-2 x)^{13/2}+\frac {17541}{176} (1-2 x)^{11/2}-\frac {39977}{144} (1-2 x)^{9/2}+\frac {13013}{32} (1-2 x)^{7/2}-\frac {41503}{160} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*(1 - 2*x)^(5/2))/160 + (13013*(1 - 2*x)^(7/2))/32 - (39977*(1 - 2*x)^(9/2))/144 + (17541*(1 - 2*x)^(11
/2))/176 - (7695*(1 - 2*x)^(13/2))/416 + (45*(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)^3 (3+5 x)^2 \, dx &=\int \left (\frac {41503}{32} (1-2 x)^{3/2}-\frac {91091}{32} (1-2 x)^{5/2}+\frac {39977}{16} (1-2 x)^{7/2}-\frac {17541}{16} (1-2 x)^{9/2}+\frac {7695}{32} (1-2 x)^{11/2}-\frac {675}{32} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {41503}{160} (1-2 x)^{5/2}+\frac {13013}{32} (1-2 x)^{7/2}-\frac {39977}{144} (1-2 x)^{9/2}+\frac {17541}{176} (1-2 x)^{11/2}-\frac {7695}{416} (1-2 x)^{13/2}+\frac {45}{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 {(1-2 x)^{5/2} \left (289575 x^5+1180575 x^4+2045655 x^3+1944575 x^2+1074070 x+307478\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6435*((1 - 2*x)^(5/2)*(307478 + 1074070*x + 1944575*x^2 + 2045655*x^3 + 1180575*x^4 + 289575*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {289575 (1-2 x)^{15/2}-3809025 (1-2 x)^{13/2}[phone] (1-2 x)^{11/2}-57167110 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-53414361 (1-2 x)^{5/2}}{205920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-53414361*(1 - 2*x)^(5/2) + 83738655*(1 - 2*x)^(7/2) - 57167110*(1 - 2*x)^(9/2) + 20522970*(1 - 2*x)^(11/2) -
3809025*(1 - 2*x)^(13/2) + 289575*(1 - 2*x)^(15/2))/205920

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fricas [A]  time = 0.93, size = 44, normalized size = 0.56 \begin {gather*} -\frac {1}{6435} \, {\left (1158300 \, x^{7} + 3564000 \, x^{6} + 3749895 \, x^{5} + 776255 \, x^{4} - 1436365 \, x^{3} - 1121793 \, x^{2} - 155842 \, x + 307478\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)^3*(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/6435*(1158300*x^7 + 3564000*x^6 + 3749895*x^5 + 776255*x^4 - 1436365*x^3 - 1121793*x^2 - 155842*x + 307478)
*sqrt(-2*x + 1)

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giac [A]  time = 0.99, size = 97, normalized size = 1.23 \begin {gather*} -\frac {45}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {7695}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {17541}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {39977}{144} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {13013}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {41503}{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)^3*(3+5*x)^2,x, algorithm="giac")

[Out]

-45/32*(2*x - 1)^7*sqrt(-2*x + 1) - 7695/416*(2*x - 1)^6*sqrt(-2*x + 1) - 17541/176*(2*x - 1)^5*sqrt(-2*x + 1)
- 39977/144*(2*x - 1)^4*sqrt(-2*x + 1) - 13013/32*(2*x - 1)^3*sqrt(-2*x + 1) - 41503/160*(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 (289575 x^{5}+1180575 x^{4}+2045655 x^{3}+1944575 x^{2}+1074070 x +307478\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)^3*(5*x+3)^2,x)

[Out]

-1/6435*(289575*x^5+1180575*x^4+2045655*x^3+1944575*x^2+1074070*x+307478)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.48, size = 55, normalized size = 0.70 \begin {gather*} \frac {45}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {7695}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {17541}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {39977}{144} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {13013}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {41503}{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)^3*(3+5*x)^2,x, algorithm="maxima")

[Out]

45/32*(-2*x + 1)^(15/2) - 7695/416*(-2*x + 1)^(13/2) + 17541/176*(-2*x + 1)^(11/2) - 39977/144*(-2*x + 1)^(9/2
) + 13013/32*(-2*x + 1)^(7/2) - 41503/160*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {13013\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {41503\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {39977\,{\left (1-2\,x\right )}^{9/2}}{144}+\frac {17541\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {7695\,{\left (1-2\,x\right )}^{13/2}}{416}+\frac {45\,{\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)^3*(5*x + 3)^2,x)

[Out]

(13013*(1 - 2*x)^(7/2))/32 - (41503*(1 - 2*x)^(5/2))/160 - (39977*(1 - 2*x)^(9/2))/144 + (17541*(1 - 2*x)^(11/
2))/176 - (7695*(1 - 2*x)^(13/2))/416 + (45*(1 - 2*x)^(15/2))/32

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sympy [A]  time = 17.03, size = 70, normalized size = 0.89 \begin {gather*} \frac {45 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {7695 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} + \frac {17541 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {39977 \left (1 - 2 x\right )^{\frac {9}{2}}}{144} + \frac {13013 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {41503 \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)**3*(3+5*x)**2,x)

[Out]

45*(1 - 2*x)**(15/2)/32 - 7695*(1 - 2*x)**(13/2)/416 + 17541*(1 - 2*x)**(11/2)/176 - 39977*(1 - 2*x)**(9/2)/14
4 + 13013*(1 - 2*x)**(7/2)/32 - 41503*(1 - 2*x)**(5/2)/160

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