3.2084 \(\int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac {2025}{704} (1-2 x)^{11/2}+\frac {1545}{32} (1-2 x)^{9/2}-\frac {159111}{448} (1-2 x)^{7/2}+\frac {121359}{80} (1-2 x)^{5/2}-\frac {832951}{192} (1-2 x)^{3/2}+\frac {381073}{32} \sqrt {1-2 x}+\frac {290521}{64 \sqrt {1-2 x}} \]

[Out]

-832951/192*(1-2*x)^(3/2)+121359/80*(1-2*x)^(5/2)-159111/448*(1-2*x)^(7/2)+1545/32*(1-2*x)^(9/2)-2025/704*(1-2
*x)^(11/2)+290521/64/(1-2*x)^(1/2)+381073/32*(1-2*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 92, 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} \[ -\frac {2025}{704} (1-2 x)^{11/2}+\frac {1545}{32} (1-2 x)^{9/2}-\frac {159111}{448} (1-2 x)^{7/2}+\frac {121359}{80} (1-2 x)^{5/2}-\frac {832951}{192} (1-2 x)^{3/2}+\frac {381073}{32} \sqrt {1-2 x}+\frac {290521}{64 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]

[Out]

290521/(64*Sqrt[1 - 2*x]) + (381073*Sqrt[1 - 2*x])/32 - (832951*(1 - 2*x)^(3/2))/192 + (121359*(1 - 2*x)^(5/2)
)/80 - (159111*(1 - 2*x)^(7/2))/448 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704

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)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {290521}{64 (1-2 x)^{3/2}}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {832951}{64} \sqrt {1-2 x}-\frac {121359}{16} (1-2 x)^{3/2}+\frac {159111}{64} (1-2 x)^{5/2}-\frac {13905}{32} (1-2 x)^{7/2}+\frac {2025}{64} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073}{32} \sqrt {1-2 x}-\frac {832951}{192} (1-2 x)^{3/2}+\frac {121359}{80} (1-2 x)^{5/2}-\frac {159111}{448} (1-2 x)^{7/2}+\frac {1545}{32} (1-2 x)^{9/2}-\frac {2025}{704} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.47 \[ \frac {-212625 x^6-1146600 x^5-2899485 x^4-4819932 x^3-6831172 x^2-15214664 x+15380984}{1155 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]

[Out]

(15380984 - 15214664*x - 6831172*x^2 - 4819932*x^3 - 2899485*x^4 - 1146600*x^5 - 212625*x^6)/(1155*Sqrt[1 - 2*
x])

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fricas [A]  time = 0.67, size = 46, normalized size = 0.50 \[ \frac {{\left (212625 \, x^{6} + 1146600 \, x^{5} + 2899485 \, x^{4} + 4819932 \, x^{3} + 6831172 \, x^{2} + 15214664 \, x - 15380984\right )} \sqrt {-2 \, x + 1}}{1155 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/1155*(212625*x^6 + 1146600*x^5 + 2899485*x^4 + 4819932*x^3 + 6831172*x^2 + 15214664*x - 15380984)*sqrt(-2*x
+ 1)/(2*x - 1)

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giac [A]  time = 1.25, size = 92, normalized size = 1.00 \[ \frac {2025}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1545}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {159111}{448} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {121359}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {832951}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {381073}{32} \, \sqrt {-2 \, x + 1} + \frac {290521}{64 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

2025/704*(2*x - 1)^5*sqrt(-2*x + 1) + 1545/32*(2*x - 1)^4*sqrt(-2*x + 1) + 159111/448*(2*x - 1)^3*sqrt(-2*x +
1) + 121359/80*(2*x - 1)^2*sqrt(-2*x + 1) - 832951/192*(-2*x + 1)^(3/2) + 381073/32*sqrt(-2*x + 1) + 290521/64
/sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \[ -\frac {212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}[phone] x -15380984}{1155 \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4*(5*x+3)^2/(-2*x+1)^(3/2),x)

[Out]

-1/1155*(212625*x^6+1146600*x^5+2899485*x^4+4819932*x^3+6831172*x^2+15214664*x-15380984)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.61, size = 64, normalized size = 0.70 \[ -\frac {2025}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1545}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {159111}{448} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {121359}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {832951}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {381073}{32} \, \sqrt {-2 \, x + 1} + \frac {290521}{64 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-2025/704*(-2*x + 1)^(11/2) + 1545/32*(-2*x + 1)^(9/2) - 159111/448*(-2*x + 1)^(7/2) + 121359/80*(-2*x + 1)^(5
/2) - 832951/192*(-2*x + 1)^(3/2) + 381073/32*sqrt(-2*x + 1) + 290521/64/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \[ \frac {290521}{64\,\sqrt {1-2\,x}}+\frac {381073\,\sqrt {1-2\,x}}{32}-\frac {832951\,{\left (1-2\,x\right )}^{3/2}}{192}+\frac {121359\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {159111\,{\left (1-2\,x\right )}^{7/2}}{448}+\frac {1545\,{\left (1-2\,x\right )}^{9/2}}{32}-\frac {2025\,{\left (1-2\,x\right )}^{11/2}}{704} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^4*(5*x + 3)^2)/(1 - 2*x)^(3/2),x)

[Out]

290521/(64*(1 - 2*x)^(1/2)) + (381073*(1 - 2*x)^(1/2))/32 - (832951*(1 - 2*x)^(3/2))/192 + (121359*(1 - 2*x)^(
5/2))/80 - (159111*(1 - 2*x)^(7/2))/448 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704

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sympy [A]  time = 41.76, size = 82, normalized size = 0.89 \[ - \frac {2025 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {1545 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {159111 \left (1 - 2 x\right )^{\frac {7}{2}}}{448} + \frac {121359 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {832951 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} + \frac {381073 \sqrt {1 - 2 x}}{32} + \frac {290521}{64 \sqrt {1 - 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(3+5*x)**2/(1-2*x)**(3/2),x)

[Out]

-2025*(1 - 2*x)**(11/2)/704 + 1545*(1 - 2*x)**(9/2)/32 - 159111*(1 - 2*x)**(7/2)/448 + 121359*(1 - 2*x)**(5/2)
/80 - 832951*(1 - 2*x)**(3/2)/192 + 381073*sqrt(1 - 2*x)/32 + 290521/(64*sqrt(1 - 2*x))

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