3.29.92 \(\int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx\)

Optimal. Leaf size=316 \[ -\text {RootSum}\left [40 \text {$\#$1}^6-40 \text {$\#$1}^5+12 \text {$\#$1}^4-4 \text {$\#$1}^3+6 \text {$\#$1}^2-10 \text {$\#$1}+5\& ,\frac {-4 \text {$\#$1}^4 \log \left (\sqrt {4 x-3}-1\right )+4 \text {$\#$1}^4 \log \left (\text {$\#$1} \left (-\sqrt {4 x-3}\right )+\text {$\#$1}+\sqrt {2 x-1}-1\right )+12 \text {$\#$1}^3 \log \left (\sqrt {4 x-3}-1\right )-12 \text {$\#$1}^3 \log \left (\text {$\#$1} \left (-\sqrt {4 x-3}\right )+\text {$\#$1}+\sqrt {2 x-1}-1\right )-12 \text {$\#$1}^2 \log \left (\sqrt {4 x-3}-1\right )+12 \text {$\#$1}^2 \log \left (\text {$\#$1} \left (-\sqrt {4 x-3}\right )+\text {$\#$1}+\sqrt {2 x-1}-1\right )+6 \text {$\#$1} \log \left (\sqrt {4 x-3}-1\right )-6 \text {$\#$1} \log \left (\text {$\#$1} \left (-\sqrt {4 x-3}\right )+\text {$\#$1}+\sqrt {2 x-1}-1\right )+\log \left (\text {$\#$1} \left (-\sqrt {4 x-3}\right )+\text {$\#$1}+\sqrt {2 x-1}-1\right )-\log \left (\sqrt {4 x-3}-1\right )}{120 \text {$\#$1}^5-100 \text {$\#$1}^4+24 \text {$\#$1}^3-6 \text {$\#$1}^2+6 \text {$\#$1}-5}\& \right ] \]

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Rubi [F]  time = 54.41, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-16*Sqrt[-21*(359 + (63*I)*Sqrt[15])*(16*46^(2/3) - 55*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)
*Sqrt[15])^(2/3))]*ArcTan[(Sqrt[21]*(359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[-1 + 2*x])/Sqrt[-16*46^(2/3) + 55*(359
+ (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3)]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[
15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3)) + (4*Sqrt[-21*(359 + (63*I)*Sqrt[15])*(32*46^(2/3) - 131*
(359 + (63*I)*Sqrt[15])^(1/3) + 2*46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3))]*ArcTan[(Sqrt[21]*(359 + (63*I)*Sqrt
[15])^(1/6)*Sqrt[-3 + 4*x])/Sqrt[-32*46^(2/3) + 131*(359 + (63*I)*Sqrt[15])^(1/3) - 2*46^(1/3)*(359 + (63*I)*S
qrt[15])^(2/3)]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3)
) - (8*Sqrt[359 + (63*I)*Sqrt[15]]*((359 + (63*I)*Sqrt[15])^(1/3)*(21*46^(2/3)*(59 + (3*I)*Sqrt[15]) + 55*Sqrt
[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/
3)*(1501 + (165*I)*Sqrt[15]))]) + 46^(1/3)*(31521 + (3465*I)*Sqrt[15] - 16*46^(1/3)*Sqrt[21*(46^(2/3)*(59 + (3
*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqr
t[15]))]) + (359 + (63*I)*Sqrt[15])^(2/3)*(2289 - 46^(1/3)*Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*
I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]))*ArcTan[(Sqrt[2
1*(-110*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) - 2*(8*46^(2/3) - Sqrt[21*(46^(
2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501
+ (165*I)*Sqrt[15]))]))] - 42*(359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[-1 + 2*x])/Sqrt[21*(110*(359 + (63*I)*Sqrt[15
])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + 2*(8*46^(2/3) + Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(3
59 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]))]])/((
11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3))*Sqrt[(46^(2/3)*(5
9 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*
I)*Sqrt[15]))*(110*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + 2*(8*46^(2/3) + Sq
rt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(
1/3)*(1501 + (165*I)*Sqrt[15]))]))]) + (8*Sqrt[359 + (63*I)*Sqrt[15]]*((359 + (63*I)*Sqrt[15])^(1/3)*(21*46^(2
/3)*(59 + (3*I)*Sqrt[15]) + 55*Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(35
9 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]) + 46^(1/3)*(31521 + (3465*I)*Sqrt[15] - 16*
46^(1/3)*Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(
2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]) + (359 + (63*I)*Sqrt[15])^(2/3)*(2289 - 46^(1/3)*Sqrt[21*(46^(2/3
)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (
165*I)*Sqrt[15]))]))*ArcTan[(Sqrt[21*(-110*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2
/3) - 2*(8*46^(2/3) - Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I
)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]))] + 42*(359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[-1 + 2*x])
/Sqrt[21*(110*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + 2*(8*46^(2/3) + Sqrt[21
*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*
(1501 + (165*I)*Sqrt[15]))]))]])/((11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*I)
*Sqrt[15])^(4/3))*Sqrt[(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[
15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))*(110*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*S
qrt[15])^(2/3) + 2*(8*46^(2/3) + Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(
359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]))]) + (Sqrt[(2*(359 + (63*I)*Sqrt[15]))/((
4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3
)*(3361 + (393*I)*Sqrt[15]))*(16*46^(2/3) + 131*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15
])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15
])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]))]*((359 + (63*I)*Sqrt[15])^(1/3)*(84*46^(2/3)*(67 + (3*I)*S
qrt[15]) + 131*Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqr
t[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]) + 2*46^(1/3)*(70581 + (8253*I)*Sqrt[15] - 16*46^(1/3)*S
qrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2
*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]) + (359 + (63*I)*Sqrt[15])^(2/3)*(14217 - 2*46^(1/3)*Sqrt[21*(4*46^(2/3)
*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 +
(393*I)*Sqrt[15]))]))*ArcTan[(Sqrt[42*(-16*46^(2/3) - 131*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*
I)*Sqrt[15])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*
I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))])] - 42*(359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[-3 + 4*x
])/Sqrt[42*(16*46^(2/3) + 131*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + Sqrt[21
*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1
/3)*(3361 + (393*I)*Sqrt[15]))])]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*
I)*Sqrt[15])^(4/3)) - (Sqrt[(2*(359 + (63*I)*Sqrt[15]))/((4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[
15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))*(16*46^(2/3) + 131*(359
+ (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])
*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]))]
*((359 + (63*I)*Sqrt[15])^(1/3)*(84*46^(2/3)*(67 + (3*I)*Sqrt[15]) + 131*Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[
15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))
]) + 2*46^(1/3)*(70581 + (8253*I)*Sqrt[15] - 16*46^(1/3)*Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*
I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]) + (359 + (63*
I)*Sqrt[15])^(2/3)*(14217 - 2*46^(1/3)*Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3)
+ 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]))*ArcTan[(Sqrt[42*(-16*46^(2/3)
- 131*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)
*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt
[15]))])] + 42*(359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[-3 + 4*x])/Sqrt[42*(16*46^(2/3) + 131*(359 + (63*I)*Sqrt[15]
)^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqr
t[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))])]])/(11776*46^(1/3)
+ 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3)) - (84*(46^(2/3)*(59 + (3*I)*Sqrt
[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]) +
(16*46^(2/3) - 55*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3))*Sqrt[(46^(2/3)*(59
+ (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)
*Sqrt[15]))/21])*Sqrt[-((359 + (63*I)*Sqrt[15])/((46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3)
+ 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))*(16*46^(2/3) + 110*(359 + (63*I)*Sq
rt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) - 2*Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*
I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))])))]*Log[Sqrt[21*
(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(
1501 + (165*I)*Sqrt[15]))] - 21*(359 + (63*I)*Sqrt[15])^(1/3)*(1 - 2*x) - (359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[2
1*(-110*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) - 2*(8*46^(2/3) - Sqrt[21*(46^(
2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501
+ (165*I)*Sqrt[15]))]))]*Sqrt[-1 + 2*x]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359
+ (63*I)*Sqrt[15])^(4/3)) + (84*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63
*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]) + (16*46^(2/3) - 55*(359 + (63*I)*Sqrt[15])^(1/3) + 4
6^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3))*Sqrt[(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 10
9*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))/21])*Sqrt[-((359 + (63*I)*Sqrt[15])/((46
^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(150
1 + (165*I)*Sqrt[15]))*(16*46^(2/3) + 110*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/
3) - 2*Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/
3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))])))]*Log[Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[1
5])^(1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))] - 21*(359 + (63*I)*Sqrt[1
5])^(1/3)*(1 - 2*x) + (359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[21*(-110*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(35
9 + (63*I)*Sqrt[15])^(2/3) - 2*(8*46^(2/3) - Sqrt[21*(46^(2/3)*(59 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(
1/3) + 109*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(1/3)*(1501 + (165*I)*Sqrt[15]))]))]*Sqrt[-1 + 2*x]])/(11776*46^
(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3)) + (21*(4*46^(2/3)*(67 + (3
*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*S
qrt[15]) + (32*46^(2/3) - 131*(359 + (63*I)*Sqrt[15])^(1/3) + 2*46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3))*Sqrt[(
4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3
)*(3361 + (393*I)*Sqrt[15]))/21])*Sqrt[-1/2*(359 + (63*I)*Sqrt[15])/((4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 +
(63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))*(16*46^(2/3
) + 131*(359 + (63*I)*Sqrt[15])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) - Sqrt[21*(4*46^(2/3)*(67 + (3*
I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sq
rt[15]))]))]*Log[Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*S
qrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))] - 21*(359 + (63*I)*Sqrt[15])^(1/3)*(3 - 4*x) - (359 +
(63*I)*Sqrt[15])^(1/6)*Sqrt[42*(-16*46^(2/3) - 131*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt
[15])^(2/3) + Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt
[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))])]*Sqrt[-3 + 4*x]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sq
rt[15])^(2/3) + 46^(2/3)*(359 + (63*I)*Sqrt[15])^(4/3)) - (21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*
Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]) + (32*46^(2/3) - 13
1*(359 + (63*I)*Sqrt[15])^(1/3) + 2*46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3))*Sqrt[(4*46^(2/3)*(67 + (3*I)*Sqrt[
15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))
/21])*Sqrt[-1/2*(359 + (63*I)*Sqrt[15])/((4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677
*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))*(16*46^(2/3) + 131*(359 + (63*I)*Sqrt[1
5])^(1/3) + 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) - Sqrt[21*(4*46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*S
qrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(3361 + (393*I)*Sqrt[15]))]))]*Log[Sqrt[21*(4*
46^(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*
(3361 + (393*I)*Sqrt[15]))] - 21*(359 + (63*I)*Sqrt[15])^(1/3)*(3 - 4*x) + (359 + (63*I)*Sqrt[15])^(1/6)*Sqrt[
42*(-16*46^(2/3) - 131*(359 + (63*I)*Sqrt[15])^(1/3) - 46^(1/3)*(359 + (63*I)*Sqrt[15])^(2/3) + Sqrt[21*(4*46^
(2/3)*(67 + (3*I)*Sqrt[15])*(359 + (63*I)*Sqrt[15])^(1/3) + 677*(359 + (63*I)*Sqrt[15])^(2/3) + 2*46^(1/3)*(33
61 + (393*I)*Sqrt[15]))])]*Sqrt[-3 + 4*x]])/(11776*46^(1/3) + 736*(359 + (63*I)*Sqrt[15])^(2/3) + 46^(2/3)*(35
9 + (63*I)*Sqrt[15])^(4/3)) + 6*Defer[Subst][Defer[Int][x^2/(9 + 109*x^2 + 55*x^4 + 7*x^6), x], x, Sqrt[-1 + 2
*x]] + 6*Defer[Subst][Defer[Int][x^4/(9 + 109*x^2 + 55*x^4 + 7*x^6), x], x, Sqrt[-1 + 2*x]] - 12*Defer[Subst][
Defer[Int][x^2/(625 + 677*x^2 + 131*x^4 + 7*x^6), x], x, Sqrt[-3 + 4*x]] - 4*Defer[Subst][Defer[Int][x^4/(625
+ 677*x^2 + 131*x^4 + 7*x^6), x], x, Sqrt[-3 + 4*x]]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx &=\int \left (\frac {4 \sqrt {-1+2 x}}{-13+10 x+34 x^2+14 x^3}+\frac {3 x \sqrt {-1+2 x}}{-13+10 x+34 x^2+14 x^3}-\frac {\sqrt {-3+4 x}}{-13+10 x+34 x^2+14 x^3}-\frac {x \sqrt {-3+4 x}}{-13+10 x+34 x^2+14 x^3}\right ) \, dx\\ &=3 \int \frac {x \sqrt {-1+2 x}}{-13+10 x+34 x^2+14 x^3} \, dx+4 \int \frac {\sqrt {-1+2 x}}{-13+10 x+34 x^2+14 x^3} \, dx-\int \frac {\sqrt {-3+4 x}}{-13+10 x+34 x^2+14 x^3} \, dx-\int \frac {x \sqrt {-3+4 x}}{-13+10 x+34 x^2+14 x^3} \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

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Mathematica [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.35, size = 176, normalized size = 0.56 \begin {gather*} 4 \text {RootSum}\left [2+3 \sqrt {2}+50 \text {$\#$1}^2+97 \sqrt {2} \text {$\#$1}^2-50 \text {$\#$1}^4+97 \sqrt {2} \text {$\#$1}^4-2 \text {$\#$1}^6+3 \sqrt {2} \text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {-3+4 x}+\sqrt {-2+4 x}-\text {$\#$1}\right )+\log \left (-\sqrt {-3+4 x}+\sqrt {-2+4 x}-\text {$\#$1}\right ) \text {$\#$1}^4}{50 \text {$\#$1}+97 \sqrt {2} \text {$\#$1}-100 \text {$\#$1}^3+194 \sqrt {2} \text {$\#$1}^3-6 \text {$\#$1}^5+9 \sqrt {2} \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + 2*x]*(4 + 3*x) + (1 + x)*Sqrt[-3 + 4*x])^(-1),x]

[Out]

4*RootSum[2 + 3*Sqrt[2] + 50*#1^2 + 97*Sqrt[2]*#1^2 - 50*#1^4 + 97*Sqrt[2]*#1^4 - 2*#1^6 + 3*Sqrt[2]*#1^6 & ,
(-Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] - #1] + Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] - #1]*#1^4)/(50*#1 + 97*Sq
rt[2]*#1 - 100*#1^3 + 194*Sqrt[2]*#1^3 - 6*#1^5 + 9*Sqrt[2]*#1^5) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x + 4\right )} \sqrt {2 \, x - 1} + \sqrt {4 \, x - 3} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 232, normalized size = 0.73

method	result	size
default	\(-8 \left (\munderset {\textit {\_R} =\RootOf \left (7 \textit {\_Z}^{6}+131 \textit {\_Z}^{4}+677 \textit {\_Z}^{2}+625\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {-3+4 x}-\textit {\_R} \right )}{21 \textit {\_R}^{5}+262 \textit {\_R}^{3}+677 \textit {\_R}}\right )+8 \left (\munderset {\textit {\_R} =\RootOf \left (7 \textit {\_Z}^{6}+55 \textit {\_Z}^{4}+109 \textit {\_Z}^{2}+9\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {-1+2 x}-\textit {\_R} \right )}{21 \textit {\_R}^{5}+110 \textit {\_R}^{3}+109 \textit {\_R}}\right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (7 \textit {\_Z}^{6}+131 \textit {\_Z}^{4}+677 \textit {\_Z}^{2}+625\right )}{\sum }\frac {\left (\textit {\_R}^{4}+3 \textit {\_R}^{2}\right ) \ln \left (\sqrt {-3+4 x}-\textit {\_R} \right )}{21 \textit {\_R}^{5}+262 \textit {\_R}^{3}+677 \textit {\_R}}\right )+3 \left (\munderset {\textit {\_R} =\RootOf \left (7 \textit {\_Z}^{6}+55 \textit {\_Z}^{4}+109 \textit {\_Z}^{2}+9\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\textit {\_R}^{2}\right ) \ln \left (\sqrt {-1+2 x}-\textit {\_R} \right )}{21 \textit {\_R}^{5}+110 \textit {\_R}^{3}+109 \textit {\_R}}\right )\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*sum(_R^2/(21*_R^5+262*_R^3+677*_R)*ln((-3+4*x)^(1/2)-_R),_R=RootOf(7*_Z^6+131*_Z^4+677*_Z^2+625))+8*sum(_R^
2/(21*_R^5+110*_R^3+109*_R)*ln((-1+2*x)^(1/2)-_R),_R=RootOf(7*_Z^6+55*_Z^4+109*_Z^2+9))-2*sum((_R^4+3*_R^2)/(2
1*_R^5+262*_R^3+677*_R)*ln((-3+4*x)^(1/2)-_R),_R=RootOf(7*_Z^6+131*_Z^4+677*_Z^2+625))+3*sum((_R^4+_R^2)/(21*_
R^5+110*_R^3+109*_R)*ln((-1+2*x)^(1/2)-_R),_R=RootOf(7*_Z^6+55*_Z^4+109*_Z^2+9))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x + 4\right )} \sqrt {2 \, x - 1} + \sqrt {4 \, x - 3} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x + 1\right ) \sqrt {4 x - 3} + \sqrt {2 x - 1} \left (3 x + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x + 1)*sqrt(4*x - 3) + sqrt(2*x - 1)*(3*x + 4)), x)

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