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3.4.18 \(\int \frac {2 b+a x^3}{\sqrt {-b+a x^3} (-b+x^2+a x^3)} \, dx\)

Optimal. Leaf size=28 \[ 2 \tan ^{-1}\left (\frac {x \sqrt {a x^3-b}}{b-a x^3}\right ) \]

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Rubi [F]  time = 1.04, 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 {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-b + x^2 + a*x^3)),x]

[Out]

(-2*Sqrt[2 - Sqrt[3]]*(b^(1/3) - a^(1/3)*x)*Sqrt[(b^(2/3) + a^(1/3)*b^(1/3)*x + a^(2/3)*x^2)/((1 - Sqrt[3])*b^
(1/3) - a^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*b^(1/3) - a^(1/3)*x)/((1 - Sqrt[3])*b^(1/3) - a^(1/3)*x)
], -7 + 4*Sqrt[3]])/(3^(1/4)*a^(1/3)*Sqrt[-((b^(1/3)*(b^(1/3) - a^(1/3)*x))/((1 - Sqrt[3])*b^(1/3) - a^(1/3)*x
)^2)]*Sqrt[-b + a*x^3]) - 3*b*Defer[Int][1/((b - x^2 - a*x^3)*Sqrt[-b + a*x^3]), x] - Defer[Int][x^2/(Sqrt[-b
+ a*x^3]*(-b + x^2 + a*x^3)), x]

Rubi steps

\begin {align*} \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b+a x^3}}+\frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-b+a x^3}} \, dx+\int \frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}+\int \left (-\frac {3 b}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}}-\frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}-(3 b) \int \frac {1}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}} \, dx-\int \frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.25, size = 2752, normalized size = 98.29 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-b + x^2 + a*x^3)),x]

[Out]

(2*Sqrt[(-(b^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/3)*b^(1/3))/a^(1/3))]*(((-1)^(1/3)*b^(1/3))/a^
(1/3) + x)*Sqrt[(-(((-1)^(2/3)*b^(1/3))/a^(1/3)) + x)/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/
a^(1/3))]*EllipticF[ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(
1/3)])/(Sqrt[(((-1)^(1/3)*b^(1/3))/a^(1/3) + x)/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))]
*Sqrt[-b + a*x^3]) + (4*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sqrt[(-(b^(1/3)/a^(
1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/a^(1/3) - x)*(((-1
)^(1/3)*b^(1/3))/a^(1/3) + x))/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[
((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#1^3 & , 1]), ArcSi
n[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sqrt[-b + a*x^3
]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 + a*#1^3 & , 1])*(Root[-b + #1^2 + a*#1^3 & , 1] - Root[-b +
#1^2 + a*#1^3 & , 2])*(Root[-b + #1^2 + a*#1^3 & , 1] - Root[-b + #1^2 + a*#1^3 & , 3])) + (2*(-(((-1)^(1/3)*b
^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*Sqrt[(-(b^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/
3)*b^(1/3))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/a^(1/3) - x)*(((-1)^(1/3)*b^(1/3))/a^(1/3) + x))/(-(((-1)^(1
/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/
((-1)^(1/3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#1^3 & , 1]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/((
(-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*Root[-b + #1^2 + a*#1^3 & , 1]^3)/(Sqrt[-b + a*x^3]*(((-1)^(1
/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 + a*#1^3 & , 1])*(Root[-b + #1^2 + a*#1^3 & , 1] - Root[-b + #1^2 + a*#1
^3 & , 2])*(Root[-b + #1^2 + a*#1^3 & , 1] - Root[-b + #1^2 + a*#1^3 & , 3])) + (4*(-(((-1)^(1/3)*b^(1/3))/a^(
1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sqrt[(-(b^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/3)*b^(1/3
))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/a^(1/3) - x)*(((-1)^(1/3)*b^(1/3))/a^(1/3) + x))/(-(((-1)^(1/3)*b^(1/
3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/
3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3
) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sqrt[-b + a*x^3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 +
a*#1^3 & , 2])*(-Root[-b + #1^2 + a*#1^3 & , 1] + Root[-b + #1^2 + a*#1^3 & , 2])*(Root[-b + #1^2 + a*#1^3 & ,
 2] - Root[-b + #1^2 + a*#1^3 & , 3])) + (2*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*S
qrt[(-(b^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/
a^(1/3) - x)*(((-1)^(1/3)*b^(1/3))/a^(1/3) + x))/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/
3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#
1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*R
oot[-b + #1^2 + a*#1^3 & , 2]^3)/(Sqrt[-b + a*x^3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 + a*#1^3 & ,
 2])*(-Root[-b + #1^2 + a*#1^3 & , 1] + Root[-b + #1^2 + a*#1^3 & , 2])*(Root[-b + #1^2 + a*#1^3 & , 2] - Root
[-b + #1^2 + a*#1^3 & , 3])) + (4*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sqrt[(-(b
^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3)) - ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/a^(1/3)
- x)*(((-1)^(1/3)*b^(1/3))/a^(1/3) + x))/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*E
llipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#1^3 & ,
3]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sqrt[
-b + a*x^3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 + a*#1^3 & , 3])*(-Root[-b + #1^2 + a*#1^3 & , 1] +
 Root[-b + #1^2 + a*#1^3 & , 3])*(-Root[-b + #1^2 + a*#1^3 & , 2] + Root[-b + #1^2 + a*#1^3 & , 3])) + (2*(-((
(-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*Sqrt[(-(b^(1/3)/a^(1/3)) + x)/(-(b^(1/3)/a^(1/3))
 - ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((((-1)^(2/3)*b^(1/3))/a^(1/3) - x)*(((-1)^(1/3)*b^(1/3))/a^(1/3) + x))
/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/
3)*b^(1/3))/((-1)^(1/3)*b^(1/3) + a^(1/3)*Root[-b + #1^2 + a*#1^3 & , 3]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) + a
^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*Root[-b + #1^2 + a*#1^3 & , 3]^3)/(Sqrt[-b + a*x^
3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-b + #1^2 + a*#1^3 & , 3])*(-Root[-b + #1^2 + a*#1^3 & , 1] + Root[-b
+ #1^2 + a*#1^3 & , 3])*(-Root[-b + #1^2 + a*#1^3 & , 2] + Root[-b + #1^2 + a*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 0.56, size = 28, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x \sqrt {a x^3-b}}{b-a x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-b + x^2 + a*x^3)),x]

[Out]

2*ArcTan[(x*Sqrt[-b + a*x^3])/(b - a*x^3)]

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fricas [B]  time = 0.44, size = 40, normalized size = 1.43 \begin {gather*} \arctan \left (\frac {{\left (a x^{3} - x^{2} - b\right )} \sqrt {a x^{3} - b}}{2 \, {\left (a x^{4} - b x\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(1/2*(a*x^3 - x^2 - b)*sqrt(a*x^3 - b)/(a*x^4 - b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^3 + 2*b)/((a*x^3 + x^2 - b)*sqrt(a*x^3 - b)), x)

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maple [C]  time = 0.52, size = 788, normalized size = 28.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*I*3^(1/2)/a*(a^2*b)^(1/3)*(-I*(x+1/2/a*(a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(a^2*b)^(1/3))*3^(1/2)*a/(a^2*b)^(1/3
))^(1/2)*((x-1/a*(a^2*b)^(1/3))/(-3/2/a*(a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(a^2*b)^(1/3)))^(1/2)*(I*(x+1/2/a*(a^2*b
)^(1/3)-1/2*I*3^(1/2)/a*(a^2*b)^(1/3))*3^(1/2)*a/(a^2*b)^(1/3))^(1/2)/(a*x^3-b)^(1/2)*EllipticF(1/3*3^(1/2)*(-
I*(x+1/2/a*(a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(a^2*b)^(1/3))*3^(1/2)*a/(a^2*b)^(1/3))^(1/2),(-I*3^(1/2)/a*(a^2*b)^(
1/3)/(-3/2/a*(a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(a^2*b)^(1/3)))^(1/2))+I/a^2/b*2^(1/2)*sum((-_alpha^2+3*b)/_alpha/(
3*_alpha*a+2)*(a^2*b)^(1/3)*(-1/2*I*a*(2*x+1/a*(I*3^(1/2)*(a^2*b)^(1/3)+(a^2*b)^(1/3)))/(a^2*b)^(1/3))^(1/2)*(
a*(x-1/a*(a^2*b)^(1/3))/(-3*(a^2*b)^(1/3)-I*3^(1/2)*(a^2*b)^(1/3)))^(1/2)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*(a^2*b
)^(1/3)+(a^2*b)^(1/3)))/(a^2*b)^(1/3))^(1/2)/(a*x^3-b)^(1/2)*(-I*(a^2*b)^(1/3)*3^(1/2)*_alpha^2*a^2+I*(a^2*b)^
(2/3)*3^(1/2)*_alpha*a-(a^2*b)^(1/3)*_alpha^2*a^2-I*(a^2*b)^(1/3)*3^(1/2)*_alpha*a-_alpha*(a^2*b)^(2/3)*a+I*(a
^2*b)^(2/3)*3^(1/2)-(a^2*b)^(1/3)*_alpha*a+2*a^2*b-(a^2*b)^(2/3))*EllipticPi(1/3*3^(1/2)*(-I*(x+1/2/a*(a^2*b)^
(1/3)+1/2*I*3^(1/2)/a*(a^2*b)^(1/3))*3^(1/2)*a/(a^2*b)^(1/3))^(1/2),1/2/a*(I*(a^2*b)^(2/3)*_alpha^2*3^(1/2)*a+
I*_alpha*3^(1/2)*a^2*b-3*(a^2*b)^(2/3)*_alpha^2*a+I*(a^2*b)^(2/3)*3^(1/2)*_alpha-2*I*(a^2*b)^(1/3)*3^(1/2)*a*b
+3*_alpha*a^2*b+I*3^(1/2)*a*b-3*(a^2*b)^(2/3)*_alpha+3*a*b)/b,(-I*3^(1/2)/a*(a^2*b)^(1/3)/(-3/2/a*(a^2*b)^(1/3
)-1/2*I*3^(1/2)/a*(a^2*b)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*a+_Z^2-b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^3 + 2*b)/((a*x^3 + x^2 - b)*sqrt(a*x^3 - b)), x)

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mupad [B]  time = 2.06, size = 45, normalized size = 1.61 \begin {gather*} \ln \left (\frac {b-a\,x^3+x^2-x\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{a\,x^3+x^2-b}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((b - a*x^3 + x^2 - x*(a*x^3 - b)^(1/2)*2i)/(a*x^3 - b + x^2))*1i

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sympy [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((a*x**3+2*b)/(a*x**3-b)**(1/2)/(a*x**3+x**2-b),x)

[Out]

Timed out

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