3.21.84 \(\int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\) [2084]

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

[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 = 0.01, 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 = {90} \begin {gather*} -\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}} \end {gather*}

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 90

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 \begin {gather*} \frac {15380984-15214664 x-6831172 x^2-4819932 x^3-2899485 x^4-1146600 x^5-212625 x^6}{1155 \sqrt {1-2 x}} \end {gather*}

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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Maple [A]
time = 0.13, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}[phone] x -15380984}{1155 \sqrt {1-2 x}}\)	\(40\)
risch	\(-\frac {212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}[phone] x -15380984}{1155 \sqrt {1-2 x}}\)	\(40\)
trager	\(\frac {\left (212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}[phone] x -15380984\right ) \sqrt {1-2 x}}{-1155+2310 x}\)	\(47\)
derivativedivides	\(-\frac {832951 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {121359 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {159111 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1545 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073 \sqrt {1-2 x}}{32}\)	\(65\)
default	\(-\frac {832951 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {121359 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {159111 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1545 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073 \sqrt {1-2 x}}{32}\)	\(65\)
meijerg	\(-\frac {144 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-1344 \sqrt {\pi }+\frac {168 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {1306 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-\frac {21648 \sqrt {\pi }}{5}+\frac {1353 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{40 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {12609 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {6960 \sqrt {\pi }}{7}+\frac {435 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{448 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {2025 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Maxima [A]
time = 0.27, size = 64, normalized size = 0.70 \begin {gather*} -\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}} \end {gather*}

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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Fricas [A]
time = 1.08, size = 46, normalized size = 0.50 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [A]
time = 22.46, size = 82, normalized size = 0.89 \begin {gather*} - \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}} \end {gather*}

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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Giac [A]
time = 1.67, size = 92, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \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} \end {gather*}

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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