3.248 \(\int \frac{B+A x}{\left (17-18 x+5 x^2\right ) \sqrt{13-22 x+10 x^2}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{(2 A+B) \tan ^{-1}\left (\frac{\sqrt{35} (2-x)}{\sqrt{10 x^2-22 x+13}}\right )}{\sqrt{35}}-\frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{35} (1-x)}{2 \sqrt{10 x^2-22 x+13}}\right )}{2 \sqrt{35}} \]

[Out]

-(((2*A + B)*ArcTan[(Sqrt[35]*(2 - x))/Sqrt[13 - 22*x + 10*x^2]])/Sqrt[35]) - ((
A + B)*ArcTanh[(Sqrt[35]*(1 - x))/(2*Sqrt[13 - 22*x + 10*x^2])])/(2*Sqrt[35])

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Rubi [A]  time = 0.288736, antiderivative size = 89, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(2 A+B) \tan ^{-1}\left (\frac{\sqrt{35} (2-x)}{\sqrt{10 x^2-22 x+13}}\right )}{\sqrt{35}}-\frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{35} (-x (A+B)+A+B)}{2 \sqrt{10 x^2-22 x+13} (A+B)}\right )}{2 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[(B + A*x)/((17 - 18*x + 5*x^2)*Sqrt[13 - 22*x + 10*x^2]),x]

[Out]

-(((2*A + B)*ArcTan[(Sqrt[35]*(2 - x))/Sqrt[13 - 22*x + 10*x^2]])/Sqrt[35]) - ((
A + B)*ArcTanh[(Sqrt[35]*(A + B - (A + B)*x))/(2*(A + B)*Sqrt[13 - 22*x + 10*x^2
])])/(2*Sqrt[35])

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Rubi in Sympy [A]  time = 22.7356, size = 105, normalized size = 1.31 \[ \frac{\sqrt{35} \left (A + B\right ) \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 140 A - 140 B + x \left (140 A + 140 B\right )\right )}{280 \left (A + B\right ) \sqrt{10 x^{2} - 22 x + 13}} \right )}}{70} - \frac{\sqrt{35} \left (2 A + B\right ) \operatorname{atan}{\left (\frac{\sqrt{35} \left (2240 A + 1120 B + x \left (- 1120 A - 560 B\right )\right )}{560 \left (2 A + B\right ) \sqrt{10 x^{2} - 22 x + 13}} \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((A*x+B)/(5*x**2-18*x+17)/(10*x**2-22*x+13)**(1/2),x)

[Out]

sqrt(35)*(A + B)*atanh(sqrt(35)*(-140*A - 140*B + x*(140*A + 140*B))/(280*(A + B
)*sqrt(10*x**2 - 22*x + 13)))/70 - sqrt(35)*(2*A + B)*atan(sqrt(35)*(2240*A + 11
20*B + x*(-1120*A - 560*B))/(560*(2*A + B)*sqrt(10*x**2 - 22*x + 13)))/35

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Mathematica [C]  time = 4.38478, size = 1149, normalized size = 14.36 \[ \frac{((8-2 i) A+(4-2 i) B) \tan ^{-1}\left (\frac{4 \left ((300+2800 i) x^3-(1900+11640 i) x^2+(3811+15444 i) x-(2494+6746 i)\right ) A^2+(4+8 i) B \left ((1240+320 i) x^3-(5354+1092 i) x^2+(7427+942 i) x-(3439+76 i)\right ) A+(2+4 i) B^2 \left ((645+110 i) x^3-(2827+336 i) x^2+(3955+186 i) x-(1843-92 i)\right )}{\left ((-4800+2800 i) x^3+10 \left ((34+17 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(1969-892 i)\right ) x^2-\left ((1054+357 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(25633-9460 i)\right ) x+(748+187 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(10987-3210 i)\right ) A^2+B \left ((-4225+4200 i) x^3+(10+5 i) \left (36 \sqrt{35} \sqrt{10 x^2-22 x+13}+(828-1871 i)\right ) x^2-\left ((1116+378 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(22801-16808 i)\right ) x+(792+198 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(9519-6362 i)\right ) A+(1+2 i) B^2 \left ((395+610 i) x^3+(4-3 i) \left (10 \sqrt{35} \sqrt{10 x^2-22 x+13}+(80-549 i)\right ) x^2+\left ((1540+3036 i)-(104-103 i) \sqrt{35} \sqrt{10 x^2-22 x+13}\right ) x+(66-77 i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(608+1208 i)\right )}\right )+((-2+8 i) A-(2-4 i) B) \tanh ^{-1}\left (\frac{\left ((-1400+12175 i) x^3+(5990-39425 i) x^2-(8096-43289 i) x+(3594-15991 i)\right ) A^2+(4+2 i) B \left ((955+3310 i) x^3-(2993+11256 i) x^2+(3185+12882 i) x-(1147+4952 i)\right ) A+(2+i) B^2 \left ((355+2110 i) x^3-(1073+7336 i) x^2+(1085+8506 i) x-(367+3288 i)\right )}{2 \left (\left ((4800+2800 i) x^3-10 i \left ((17+34 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(892-1969 i)\right ) x^2+\left ((25633+9460 i)-(1054-357 i) \sqrt{35} \sqrt{10 x^2-22 x+13}\right ) x+(1-4 i) \left ((88+165 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(109-2774 i)\right )\right ) A^2+B \left ((4225+4200 i) x^3+(10-5 i) \left (36 \sqrt{35} \sqrt{10 x^2-22 x+13}-(828+1871 i)\right ) x^2+\left ((22801+16808 i)-(1116-378 i) \sqrt{35} \sqrt{10 x^2-22 x+13}\right ) x+(792-198 i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(9519+6362 i)\right ) A+(2+i) B^2 \left ((610+395 i) x^3+(3-4 i) \left (10 \sqrt{35} \sqrt{10 x^2-22 x+13}-(80+549 i)\right ) x^2+\left ((3036+1540 i)-(103-104 i) \sqrt{35} \sqrt{10 x^2-22 x+13}\right ) x+(77-66 i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(1208+608 i)\right )\right )}\right )-2 A \log \left (i \left (5 x^2-18 x+17\right )\right )-2 B \log \left (i \left (5 x^2-18 x+17\right )\right )+(1-4 i) A \log \left ((1+2 i) \left ((-25-1350 i) x^2-70 i \sqrt{35} \sqrt{10 x^2-22 x+13} x+(118+2844 i) x+68 i \sqrt{35} \sqrt{10 x^2-22 x+13}-(127+1566 i)\right )\right )+(1-2 i) B \log \left ((1+2 i) \left ((-25-1350 i) x^2-70 i \sqrt{35} \sqrt{10 x^2-22 x+13} x+(118+2844 i) x+68 i \sqrt{35} \sqrt{10 x^2-22 x+13}-(127+1566 i)\right )\right )+(1+4 i) A \log \left ((2+i) \left ((1350+25 i) x^2+70 \sqrt{35} \sqrt{10 x^2-22 x+13} x-(2844+118 i) x-68 \sqrt{35} \sqrt{10 x^2-22 x+13}+(1566+127 i)\right )\right )+(1+2 i) B \log \left ((2+i) \left ((1350+25 i) x^2+70 \sqrt{35} \sqrt{10 x^2-22 x+13} x-(2844+118 i) x-68 \sqrt{35} \sqrt{10 x^2-22 x+13}+(1566+127 i)\right )\right )}{8 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Integrate[(B + A*x)/((17 - 18*x + 5*x^2)*Sqrt[13 - 22*x + 10*x^2]),x]

[Out]

(((8 - 2*I)*A + (4 - 2*I)*B)*ArcTan[(4*A^2*((-2494 - 6746*I) + (3811 + 15444*I)*
x - (1900 + 11640*I)*x^2 + (300 + 2800*I)*x^3) + (2 + 4*I)*B^2*((-1843 + 92*I) +
(3955 + 186*I)*x - (2827 + 336*I)*x^2 + (645 + 110*I)*x^3) + (4 + 8*I)*A*B*((-3
439 - 76*I) + (7427 + 942*I)*x - (5354 + 1092*I)*x^2 + (1240 + 320*I)*x^3))/((1
+ 2*I)*B^2*((-608 - 1208*I) + (395 + 610*I)*x^3 + (66 - 77*I)*Sqrt[35]*Sqrt[13 -
22*x + 10*x^2] + x*((1540 + 3036*I) - (104 - 103*I)*Sqrt[35]*Sqrt[13 - 22*x + 1
0*x^2]) + (4 - 3*I)*x^2*((80 - 549*I) + 10*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2])) +
A^2*((10987 - 3210*I) - (4800 - 2800*I)*x^3 + (748 + 187*I)*Sqrt[35]*Sqrt[13 -
22*x + 10*x^2] + 10*x^2*((1969 - 892*I) + (34 + 17*I)*Sqrt[35]*Sqrt[13 - 22*x +
10*x^2]) - x*((25633 - 9460*I) + (1054 + 357*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2
])) + A*B*((9519 - 6362*I) - (4225 - 4200*I)*x^3 + (792 + 198*I)*Sqrt[35]*Sqrt[1
3 - 22*x + 10*x^2] + (10 + 5*I)*x^2*((828 - 1871*I) + 36*Sqrt[35]*Sqrt[13 - 22*x
+ 10*x^2]) - x*((22801 - 16808*I) + (1116 + 378*I)*Sqrt[35]*Sqrt[13 - 22*x + 10
*x^2])))] + ((-2 + 8*I)*A - (2 - 4*I)*B)*ArcTanh[(A^2*((3594 - 15991*I) - (8096
- 43289*I)*x + (5990 - 39425*I)*x^2 - (1400 - 12175*I)*x^3) + (2 + I)*B^2*((-367
- 3288*I) + (1085 + 8506*I)*x - (1073 + 7336*I)*x^2 + (355 + 2110*I)*x^3) + (4
+ 2*I)*A*B*((-1147 - 4952*I) + (3185 + 12882*I)*x - (2993 + 11256*I)*x^2 + (955
+ 3310*I)*x^3))/(2*((2 + I)*B^2*((-1208 - 608*I) + (610 + 395*I)*x^3 + (77 - 66*
I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2] + x*((3036 + 1540*I) - (103 - 104*I)*Sqrt[3
5]*Sqrt[13 - 22*x + 10*x^2]) + (3 - 4*I)*x^2*((-80 - 549*I) + 10*Sqrt[35]*Sqrt[1
3 - 22*x + 10*x^2])) + A*B*((-9519 - 6362*I) + (4225 + 4200*I)*x^3 + (792 - 198*
I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2] + x*((22801 + 16808*I) - (1116 - 378*I)*Sqr
t[35]*Sqrt[13 - 22*x + 10*x^2]) + (10 - 5*I)*x^2*((-828 - 1871*I) + 36*Sqrt[35]*
Sqrt[13 - 22*x + 10*x^2])) + A^2*((4800 + 2800*I)*x^3 + x*((25633 + 9460*I) - (1
054 - 357*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]) - (10*I)*x^2*((892 - 1969*I) + (
17 + 34*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]) + (1 - 4*I)*((109 - 2774*I) + (88
+ 165*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]))))] - 2*A*Log[I*(17 - 18*x + 5*x^2)]
- 2*B*Log[I*(17 - 18*x + 5*x^2)] + (1 - 4*I)*A*Log[(1 + 2*I)*((-127 - 1566*I) +
(118 + 2844*I)*x - (25 + 1350*I)*x^2 + (68*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]
- (70*I)*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])] + (1 - 2*I)*B*Log[(1 + 2*I)*((-1
27 - 1566*I) + (118 + 2844*I)*x - (25 + 1350*I)*x^2 + (68*I)*Sqrt[35]*Sqrt[13 -
22*x + 10*x^2] - (70*I)*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])] + (1 + 4*I)*A*Log[
(2 + I)*((1566 + 127*I) - (2844 + 118*I)*x + (1350 + 25*I)*x^2 - 68*Sqrt[35]*Sqr
t[13 - 22*x + 10*x^2] + 70*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])] + (1 + 2*I)*B*L
og[(2 + I)*((1566 + 127*I) - (2844 + 118*I)*x + (1350 + 25*I)*x^2 - 68*Sqrt[35]*
Sqrt[13 - 22*x + 10*x^2] + 70*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])])/(8*Sqrt[35]
)

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Maple [B]  time = 0.039, size = 192, normalized size = 2.4 \[{\frac{\sqrt{35}}{70}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9} \left ({\it Artanh} \left ({\frac{2\,\sqrt{35}}{35}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}} \right ) A-4\,\arctan \left ({\frac{\sqrt{35} \left ( -2+x \right ) }{1-x}{\frac{1}{\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}}}} \right ) A+{\it Artanh} \left ({\frac{2\,\sqrt{35}}{35}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}} \right ) B-2\,\arctan \left ({\frac{\sqrt{35} \left ( -2+x \right ) }{1-x}{\frac{1}{\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}}}} \right ) B \right ){\frac{1}{\sqrt{{1 \left ({\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9 \right ) \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((A*x+B)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x)

[Out]

1/70*((-2+x)^2/(1-x)^2+9)^(1/2)*35^(1/2)*(arctanh(2/35*((-2+x)^2/(1-x)^2+9)^(1/2
)*35^(1/2))*A-4*arctan(35^(1/2)/((-2+x)^2/(1-x)^2+9)^(1/2)*(-2+x)/(1-x))*A+arcta
nh(2/35*((-2+x)^2/(1-x)^2+9)^(1/2)*35^(1/2))*B-2*arctan(35^(1/2)/((-2+x)^2/(1-x)
^2+9)^(1/2)*(-2+x)/(1-x))*B)/(((-2+x)^2/(1-x)^2+9)/(1+(-2+x)/(1-x))^2)^(1/2)/(1+
(-2+x)/(1-x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A x + B}{\sqrt{10 \, x^{2} - 22 \, x + 13}{\left (5 \, x^{2} - 18 \, x + 17\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A*x + B)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)),x, algorithm="maxima")

[Out]

integrate((A*x + B)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A*x + B)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A x + B}{\left (5 x^{2} - 18 x + 17\right ) \sqrt{10 x^{2} - 22 x + 13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A*x+B)/(5*x**2-18*x+17)/(10*x**2-22*x+13)**(1/2),x)

[Out]

Integral((A*x + B)/((5*x**2 - 18*x + 17)*sqrt(10*x**2 - 22*x + 13)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A*x + B)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError