3.1871 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2 \, dx\)

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

[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

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Rubi [A]  time = 0.0149522, antiderivative size = 79, normalized size of antiderivative = 1., 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} \[ \frac{45}{32} (1-2 x)^{15/2}-\frac{7695}{416} (1-2 x)^{13/2}+\frac{17541}{176} (1-2 x)^{11/2}-\frac{39977}{144} (1-2 x)^{9/2}+\frac{13013}{32} (1-2 x)^{7/2}-\frac{41503}{160} (1-2 x)^{5/2} \]

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*}

Mathematica [A]  time = 0.0172434, size = 38, normalized size = 0.48 \[ -\frac{(1-2 x)^{5/2} \left (289575 x^5+1180575 x^4+2045655 x^3+1944575 x^2+1074070 x+307478\right )}{6435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{289575\,{x}^{5}+1180575\,{x}^{4}+2045655\,{x}^{3}+1944575\,{x}^{2}+1074070\,x+307478}{6435} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07087, size = 74, normalized size = 0.94 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.28566, size = 171, normalized size = 2.16 \begin{align*} -\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{align*}

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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Sympy [A]  time = 12.7484, size = 70, normalized size = 0.89 \begin{align*} \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{align*}

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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Giac [A]  time = 1.85602, size = 131, normalized size = 1.66 \begin{align*} -\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{align*}

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)