3.2085 \(\int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac {75}{32} (1-2 x)^{9/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {91091}{32} \sqrt {1-2 x}+\frac {41503}{32 \sqrt {1-2 x}} \]

[Out]

-39977/48*(1-2*x)^(3/2)+17541/80*(1-2*x)^(5/2)-7695/224*(1-2*x)^(7/2)+75/32*(1-2*x)^(9/2)+41503/32/(1-2*x)^(1/
2)+91091/32*(1-2*x)^(1/2)

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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} \[ \frac {75}{32} (1-2 x)^{9/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {91091}{32} \sqrt {1-2 x}+\frac {41503}{32 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))/80
- (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/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 \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {41503}{32 (1-2 x)^{3/2}}-\frac {91091}{32 \sqrt {1-2 x}}+\frac {39977}{16} \sqrt {1-2 x}-\frac {17541}{16} (1-2 x)^{3/2}+\frac {7695}{32} (1-2 x)^{5/2}-\frac {675}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {41503}{32 \sqrt {1-2 x}}+\frac {91091}{32} \sqrt {1-2 x}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {75}{32} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \[ \frac {-7875 x^5-38025 x^4-88443 x^3-150253 x^2-359726 x+367286}{105 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(367286 - 359726*x - 150253*x^2 - 88443*x^3 - 38025*x^4 - 7875*x^5)/(105*Sqrt[1 - 2*x])

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fricas [A]  time = 0.90, size = 41, normalized size = 0.52 \[ \frac {{\left (7875 \, x^{5} + 38025 \, x^{4} + 88443 \, x^{3} + 150253 \, x^{2} + 359726 \, x - 367286\right )} \sqrt {-2 \, x + 1}}{105 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(7875*x^5 + 38025*x^4 + 88443*x^3 + 150253*x^2 + 359726*x - 367286)*sqrt(-2*x + 1)/(2*x - 1)

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giac [A]  time = 1.24, size = 76, normalized size = 0.96 \[ \frac {75}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {7695}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {17541}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {39977}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {91091}{32} \, \sqrt {-2 \, x + 1} + \frac {41503}{32 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/32*(2*x - 1)^4*sqrt(-2*x + 1) + 7695/224*(2*x - 1)^3*sqrt(-2*x + 1) + 17541/80*(2*x - 1)^2*sqrt(-2*x + 1) -
39977/48*(-2*x + 1)^(3/2) + 91091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \[ -\frac {7875 x^{5}+38025 x^{4}+88443 x^{3}+150253 x^{2}+359726 x -367286}{105 \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(7875*x^5+38025*x^4+88443*x^3+150253*x^2+359726*x-367286)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.48, size = 55, normalized size = 0.70 \[ \frac {75}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {7695}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {17541}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {39977}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {91091}{32} \, \sqrt {-2 \, x + 1} + \frac {41503}{32 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/32*(-2*x + 1)^(9/2) - 7695/224*(-2*x + 1)^(7/2) + 17541/80*(-2*x + 1)^(5/2) - 39977/48*(-2*x + 1)^(3/2) + 9
1091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \[ \frac {41503}{32\,\sqrt {1-2\,x}}+\frac {91091\,\sqrt {1-2\,x}}{32}-\frac {39977\,{\left (1-2\,x\right )}^{3/2}}{48}+\frac {17541\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {7695\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {75\,{\left (1-2\,x\right )}^{9/2}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

41503/(32*(1 - 2*x)^(1/2)) + (91091*(1 - 2*x)^(1/2))/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))
/80 - (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/32

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sympy [A]  time = 32.85, size = 70, normalized size = 0.89 \[ \frac {75 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {7695 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {17541 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {39977 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} + \frac {91091 \sqrt {1 - 2 x}}{32} + \frac {41503}{32 \sqrt {1 - 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*(1 - 2*x)**(9/2)/32 - 7695*(1 - 2*x)**(7/2)/224 + 17541*(1 - 2*x)**(5/2)/80 - 39977*(1 - 2*x)**(3/2)/48 + 9
1091*sqrt(1 - 2*x)/32 + 41503/(32*sqrt(1 - 2*x))

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