3.2014 \(\int \frac{(2+3 x)^3 (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=79 \[ \frac{675}{352} (1-2 x)^{11/2}-\frac{855}{32} (1-2 x)^{9/2}+\frac{17541}{112} (1-2 x)^{7/2}-\frac{39977}{80} (1-2 x)^{5/2}+\frac{91091}{96} (1-2 x)^{3/2}-\frac{41503}{32} \sqrt{1-2 x} \]

[Out]

(-41503*Sqrt[1 - 2*x])/32 + (91091*(1 - 2*x)^(3/2))/96 - (39977*(1 - 2*x)^(5/2))/80 + (17541*(1 - 2*x)^(7/2))/
112 - (855*(1 - 2*x)^(9/2))/32 + (675*(1 - 2*x)^(11/2))/352

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Rubi [A]  time = 0.0145525, 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} \[ \frac{675}{352} (1-2 x)^{11/2}-\frac{855}{32} (1-2 x)^{9/2}+\frac{17541}{112} (1-2 x)^{7/2}-\frac{39977}{80} (1-2 x)^{5/2}+\frac{91091}{96} (1-2 x)^{3/2}-\frac{41503}{32} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*Sqrt[1 - 2*x])/32 + (91091*(1 - 2*x)^(3/2))/96 - (39977*(1 - 2*x)^(5/2))/80 + (17541*(1 - 2*x)^(7/2))/
112 - (855*(1 - 2*x)^(9/2))/32 + (675*(1 - 2*x)^(11/2))/352

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}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{41503}{32 \sqrt{1-2 x}}-\frac{91091}{32} \sqrt{1-2 x}+\frac{39977}{16} (1-2 x)^{3/2}-\frac{17541}{16} (1-2 x)^{5/2}+\frac{7695}{32} (1-2 x)^{7/2}-\frac{675}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{41503}{32} \sqrt{1-2 x}+\frac{91091}{96} (1-2 x)^{3/2}-\frac{39977}{80} (1-2 x)^{5/2}+\frac{17541}{112} (1-2 x)^{7/2}-\frac{855}{32} (1-2 x)^{9/2}+\frac{675}{352} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0140756, size = 38, normalized size = 0.48 \[ -\frac{\sqrt{1-2 x} \left (70875 x^5+316575 x^4+636795 x^3+790023 x^2+743822 x+826982\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(826982 + 743822*x + 790023*x^2 + 636795*x^3 + 316575*x^4 + 70875*x^5))/1155

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{70875\,{x}^{5}+316575\,{x}^{4}+636795\,{x}^{3}+790023\,{x}^{2}+743822\,x+826982}{1155}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(70875*x^5+316575*x^4+636795*x^3+790023*x^2+743822*x+826982)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.06339, size = 74, normalized size = 0.94 \begin{align*} \frac{675}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{855}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{17541}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{39977}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{91091}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{41503}{32} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675/352*(-2*x + 1)^(11/2) - 855/32*(-2*x + 1)^(9/2) + 17541/112*(-2*x + 1)^(7/2) - 39977/80*(-2*x + 1)^(5/2) +
91091/96*(-2*x + 1)^(3/2) - 41503/32*sqrt(-2*x + 1)

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Fricas [A]  time = 1.32522, size = 128, normalized size = 1.62 \begin{align*} -\frac{1}{1155} \,{\left (70875 \, x^{5} + 316575 \, x^{4} + 636795 \, x^{3} + 790023 \, x^{2} + 743822 \, x + 826982\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(70875*x^5 + 316575*x^4 + 636795*x^3 + 790023*x^2 + 743822*x + 826982)*sqrt(-2*x + 1)

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Sympy [A]  time = 42.0325, size = 70, normalized size = 0.89 \begin{align*} \frac{675 \left (1 - 2 x\right )^{\frac{11}{2}}}{352} - \frac{855 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} + \frac{17541 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} - \frac{39977 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} + \frac{91091 \left (1 - 2 x\right )^{\frac{3}{2}}}{96} - \frac{41503 \sqrt{1 - 2 x}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675*(1 - 2*x)**(11/2)/352 - 855*(1 - 2*x)**(9/2)/32 + 17541*(1 - 2*x)**(7/2)/112 - 39977*(1 - 2*x)**(5/2)/80 +
91091*(1 - 2*x)**(3/2)/96 - 41503*sqrt(1 - 2*x)/32

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Giac [A]  time = 1.70734, size = 112, normalized size = 1.42 \begin{align*} -\frac{675}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{855}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{17541}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{39977}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{91091}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{41503}{32} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-675/352*(2*x - 1)^5*sqrt(-2*x + 1) - 855/32*(2*x - 1)^4*sqrt(-2*x + 1) - 17541/112*(2*x - 1)^3*sqrt(-2*x + 1)
- 39977/80*(2*x - 1)^2*sqrt(-2*x + 1) + 91091/96*(-2*x + 1)^(3/2) - 41503/32*sqrt(-2*x + 1)