∫ (−2c+ax3)√ _______ c+bx2+ax3 _____________________________________

3.8.79 $\int \frac {(-2 c+a x^3) \sqrt {c+b x^2+a x^3}}{(c+a x^3)^2} \, dx$

Optimal. Leaf size=60 \[ -\frac {x \sqrt {a x^3+b x^2+c}}{a x^3+c}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x^3+b x^2+c}}\right )}{\sqrt {b}} \]

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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

\begin {align*} \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx &=\int \left (-\frac {3 c \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2}+\frac {\sqrt {c+b x^2+a x^3}}{c+a x^3}\right ) \, dx\\ &=-\left ((3 c) \int \frac {\sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx\right )+\int \frac {\sqrt {c+b x^2+a x^3}}{c+a x^3} \, dx \end {align*}

rest of steps removed due to Latex formating problem.

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Mathematica [C] time = 4.09, size = 1788, normalized size = 29.80

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-((x*(c + x^2*(b + a*x)))/(c + a*x^3)) + (EllipticF[ArcSin[Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 3])/(-Root

[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3])]], (Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c +

b*#1^2 + a*#1^3 & , 3])/(Root[c + b*#1^2 + a*#1^3 & , 1] - Root[c + b*#1^2 + a*#1^3 & , 3])]*(x - Root[c + b*#

1^2 + a*#1^3 & , 3])*Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 1])/(Root[c + b*#1^2 + a*#1^3 & , 1] - Root[c + b

*#1^2 + a*#1^3 & , 3])]*Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 2])/(Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c

+ b*#1^2 + a*#1^3 & , 3])])/Sqrt[(x - Root[c + b*#1^2 + a*#1^3 & , 3])/(Root[c + b*#1^2 + a*#1^3 & , 2] - Root

[c + b*#1^2 + a*#1^3 & , 3])] + (3*(-1)^(2/3)*c^(1/3)*EllipticPi[(a^(1/3)*(-Root[c + b*#1^2 + a*#1^3 & , 2] +

Root[c + b*#1^2 + a*#1^3 & , 3]))/(-((-1)^(1/3)*c^(1/3)) + a^(1/3)*Root[c + b*#1^2 + a*#1^3 & , 3]), ArcSin[Sq

rt[(-x + Root[c + b*#1^2 + a*#1^3 & , 3])/(-Root[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3])

]], (Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c + b*#1^2 + a*#1^3 & , 3])/(Root[c + b*#1^2 + a*#1^3 & , 1] - Roo

t[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 1])/(Root[c + b*#1^2 + a*#1^3 & , 1] -

Root[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[-(((x - Root[c + b*#1^2 + a*#1^3 & , 2])*(x - Root[c + b*#1^2 + a*#1^3

& , 3]))/(Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c + b*#1^2 + a*#1^3 & , 3])^2)]*(-Root[c + b*#1^2 + a*#1^3 &

, 2] + Root[c + b*#1^2 + a*#1^3 & , 3]))/((1 + (-1)^(1/3))^2*((-1)^(1/3)*c^(1/3) - a^(1/3)*Root[c + b*#1^2 + a

*#1^3 & , 3])) + (c^(1/3)*EllipticPi[(a^(1/3)*(-Root[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & ,

3]))/(c^(1/3) + a^(1/3)*Root[c + b*#1^2 + a*#1^3 & , 3]), ArcSin[Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 3])/

(-Root[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3])]], (Root[c + b*#1^2 + a*#1^3 & , 2] - Roo

t[c + b*#1^2 + a*#1^3 & , 3])/(Root[c + b*#1^2 + a*#1^3 & , 1] - Root[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[(-x +

Root[c + b*#1^2 + a*#1^3 & , 1])/(Root[c + b*#1^2 + a*#1^3 & , 1] - Root[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[-((

(x - Root[c + b*#1^2 + a*#1^3 & , 2])*(x - Root[c + b*#1^2 + a*#1^3 & , 3]))/(Root[c + b*#1^2 + a*#1^3 & , 2]

- Root[c + b*#1^2 + a*#1^3 & , 3])^2)]*(-Root[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3]))/(

c^(1/3) + a^(1/3)*Root[c + b*#1^2 + a*#1^3 & , 3]) + (2*(-1)^(2/3)*c^(1/3)*EllipticPi[(a^(1/3)*(-Root[c + b*#1

^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3]))/((-1)^(2/3)*c^(1/3) + a^(1/3)*Root[c + b*#1^2 + a*#1^3

& , 3]), ArcSin[Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 3])/(-Root[c + b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^

2 + a*#1^3 & , 3])]], (Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c + b*#1^2 + a*#1^3 & , 3])/(Root[c + b*#1^2 + a

*#1^3 & , 1] - Root[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[(-x + Root[c + b*#1^2 + a*#1^3 & , 1])/(Root[c + b*#1^2

+ a*#1^3 & , 1] - Root[c + b*#1^2 + a*#1^3 & , 3])]*Sqrt[-(((x - Root[c + b*#1^2 + a*#1^3 & , 2])*(x - Root[c

+ b*#1^2 + a*#1^3 & , 3]))/(Root[c + b*#1^2 + a*#1^3 & , 2] - Root[c + b*#1^2 + a*#1^3 & , 3])^2)]*(-Root[c +

b*#1^2 + a*#1^3 & , 2] + Root[c + b*#1^2 + a*#1^3 & , 3]))/(I*(I + Sqrt[3])*c^(1/3) + 2*a^(1/3)*Root[c + b*#1^

2 + a*#1^3 & , 3]))/Sqrt[c + x^2*(b + a*x)]

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IntegrateAlgebraic [A] time = 0.49, size = 60, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {c+b x^2+a x^3}}{c+a x^3}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {c+b x^2+a x^3}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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fricas [A] time = 0.51, size = 244, normalized size = 4.07 \begin {gather*} \left [-\frac {4 \, \sqrt {a x^{3} + b x^{2} + c} b x - {\left (a x^{3} + c\right )} \sqrt {b} \log \left (\frac {a^{2} x^{6} + 8 \, a b x^{5} + 8 \, b^{2} x^{4} + 2 \, a c x^{3} + 8 \, b c x^{2} - 4 \, {\left (a x^{4} + 2 \, b x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} \sqrt {b} + c^{2}}{a^{2} x^{6} + 2 \, a c x^{3} + c^{2}}\right )}{4 \, {\left (a b x^{3} + b c\right )}}, -\frac {2 \, \sqrt {a x^{3} + b x^{2} + c} b x - {\left (a x^{3} + c\right )} \sqrt {-b} \arctan \left (\frac {{\left (a x^{3} + 2 \, b x^{2} + c\right )} \sqrt {a x^{3} + b x^{2} + c} \sqrt {-b}}{2 \, {\left (a b x^{4} + b^{2} x^{3} + b c x\right )}}\right )}{2 \, {\left (a b x^{3} + b c\right )}}\right ] \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*sqrt(a*x^3 + b*x^2 + c)*b*x - (a*x^3 + c)*sqrt(b)*log((a^2*x^6 + 8*a*b*x^5 + 8*b^2*x^4 + 2*a*c*x^3 +

8*b*c*x^2 - 4*(a*x^4 + 2*b*x^3 + c*x)*sqrt(a*x^3 + b*x^2 + c)*sqrt(b) + c^2)/(a^2*x^6 + 2*a*c*x^3 + c^2)))/(a*

b*x^3 + b*c), -1/2*(2*sqrt(a*x^3 + b*x^2 + c)*b*x - (a*x^3 + c)*sqrt(-b)*arctan(1/2*(a*x^3 + 2*b*x^2 + c)*sqrt

(a*x^3 + b*x^2 + c)*sqrt(-b)/(a*b*x^4 + b^2*x^3 + b*c*x)))/(a*b*x^3 + b*c)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C] time = 2.38, size = 5094, normalized size = 84.90


method	result	size

elliptic	$\text {Expression too large to display}$	$5094$
default	$\text {Expression too large to display}$	$10183$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B] time = 1.47, size = 79, normalized size = 1.32 \begin {gather*} \frac {\ln \left (\frac {c+a\,x^3+2\,b\,x^2-2\,\sqrt {b}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c^2+a\,c\,x^3}\right )}{2\,\sqrt {b}}-\frac {x\,\sqrt {a\,x^3+b\,x^2+c}}{a\,x^3+c} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

+ b*x^2)^(1/2))/(c + a*x^3)

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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*c)*(a*x**3+b*x**2+c)**(1/2)/(a*x**3+c)**2,x)

[Out]

Timed out

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