3.18.97 \(\int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} (5 c+4 b x+4 a x^2)} \, dx\)

Optimal. Leaf size=122 \[ -\frac {\tan ^{-1}\left (1-\frac {2 \sqrt [4]{a x^2+b x+c}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a x^2+b x+c}}{\sqrt [4]{c}}+1\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {a x^2+b x+c}}{\sqrt [4]{c}}+\frac {\sqrt [4]{c}}{2}}{\sqrt [4]{a x^2+b x+c}}\right )}{2 \sqrt [4]{c}} \]

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Rubi [F]  time = 0.03, 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 {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x]

[Out]

Defer[Int][(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)), x]

Rubi steps

\begin {align*} \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx &=\int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x]

[Out]

Integrate[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)), x]

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IntegrateAlgebraic [A]  time = 0.30, size = 122, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (1-\frac {2 \sqrt [4]{c+b x+a x^2}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}+\frac {\tan ^{-1}\left (1+\frac {2 \sqrt [4]{c+b x+a x^2}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{2}+\frac {\sqrt {c+b x+a x^2}}{\sqrt [4]{c}}}{\sqrt [4]{c+b x+a x^2}}\right )}{2 \sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x]

[Out]

-1/2*ArcTan[1 - (2*(c + b*x + a*x^2)^(1/4))/c^(1/4)]/c^(1/4) + ArcTan[1 + (2*(c + b*x + a*x^2)^(1/4))/c^(1/4)]
/(2*c^(1/4)) - ArcTanh[(c^(1/4)/2 + Sqrt[c + b*x + a*x^2]/c^(1/4))/(c + b*x + a*x^2)^(1/4)]/(2*c^(1/4))

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fricas [A]  time = 0.50, size = 154, normalized size = 1.26 \begin {gather*} -2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \arctan \left (2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{2} \, c \sqrt {-\frac {1}{c}} + \sqrt {a x^{2} + b x + c}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \log \left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} c \left (-\frac {1}{c}\right )^{\frac {3}{4}} + {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} c \left (-\frac {1}{c}\right )^{\frac {3}{4}} + {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x+b)/(a*x^2+b*x+c)^(1/4)/(4*a*x^2+4*b*x+5*c),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.41, size = 202, normalized size = 1.66 \begin {gather*} \frac {4^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right )}}{c^{\frac {1}{4}}}\right )}{8 \, c^{\frac {1}{4}}} + \frac {4^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right )}}{c^{\frac {1}{4}}}\right )}{8 \, c^{\frac {1}{4}}} - \frac {4^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} c^{\frac {1}{4}} + \sqrt {a x^{2} + b x + c} + \frac {1}{2} \, \sqrt {c}\right )}{16 \, c^{\frac {1}{4}}} + \frac {4^{\frac {3}{4}} \sqrt {2} \log \left (-\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} c^{\frac {1}{4}} + \sqrt {a x^{2} + b x + c} + \frac {1}{2} \, \sqrt {c}\right )}{16 \, c^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x+b)/(a*x^2+b*x+c)^(1/4)/(4*a*x^2+4*b*x+5*c),x, algorithm="giac")

[Out]

1/8*4^(3/4)*sqrt(2)*arctan(2*sqrt(2)*(1/4)^(3/4)*(sqrt(2)*(1/4)^(1/4)*c^(1/4) + 2*(a*x^2 + b*x + c)^(1/4))/c^(
1/4))/c^(1/4) + 1/8*4^(3/4)*sqrt(2)*arctan(-2*sqrt(2)*(1/4)^(3/4)*(sqrt(2)*(1/4)^(1/4)*c^(1/4) - 2*(a*x^2 + b*
x + c)^(1/4))/c^(1/4))/c^(1/4) - 1/16*4^(3/4)*sqrt(2)*log(sqrt(2)*(1/4)^(1/4)*(a*x^2 + b*x + c)^(1/4)*c^(1/4)
+ sqrt(a*x^2 + b*x + c) + 1/2*sqrt(c))/c^(1/4) + 1/16*4^(3/4)*sqrt(2)*log(-sqrt(2)*(1/4)^(1/4)*(a*x^2 + b*x +
c)^(1/4)*c^(1/4) + sqrt(a*x^2 + b*x + c) + 1/2*sqrt(c))/c^(1/4)

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maple [F]  time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {2 a x +b}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}} \left (4 a \,x^{2}+4 b x +5 c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x+b)/(a*x^2+b*x+c)^(1/4)/(4*a*x^2+4*b*x+5*c),x)

[Out]

int((2*a*x+b)/(a*x^2+b*x+c)^(1/4)/(4*a*x^2+4*b*x+5*c),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x + b}{{\left (4 \, a x^{2} + 4 \, b x + 5 \, c\right )} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x+b)/(a*x^2+b*x+c)^(1/4)/(4*a*x^2+4*b*x+5*c),x, algorithm="maxima")

[Out]

integrate((2*a*x + b)/((4*a*x^2 + 4*b*x + 5*c)*(a*x^2 + b*x + c)^(1/4)), x)

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mupad [B]  time = 1.33, size = 57, normalized size = 0.47 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (a\,x^2+b\,x+c\right )}^{1/4}}{{\left (-c\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {\sqrt {2}\,{\left (a\,x^2+b\,x+c\right )}^{1/4}}{{\left (-c\right )}^{1/4}}\right )\right )}{2\,{\left (-c\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x)

[Out]

(2^(1/2)*(atan((2^(1/2)*(c + b*x + a*x^2)^(1/4))/(-c)^(1/4)) - atanh((2^(1/2)*(c + b*x + a*x^2)^(1/4))/(-c)^(1
/4))))/(2*(-c)^(1/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a x + b}{\sqrt [4]{a x^{2} + b x + c} \left (4 a x^{2} + 4 b x + 5 c\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x+b)/(a*x**2+b*x+c)**(1/4)/(4*a*x**2+4*b*x+5*c),x)

[Out]

Integral((2*a*x + b)/((a*x**2 + b*x + c)**(1/4)*(4*a*x**2 + 4*b*x + 5*c)), x)

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