Optimal. Leaf size=239 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}} \]
________________________________________________________________________________________
Rubi [F] time = 3.67, 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^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-b^5+a^5 x^5}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^5+a^5 x^5\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-b^5+a^5 x^{10}}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}-\frac {2 b^5}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{5 b^4 \left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {4 b^3-3 a b^2 x^2+2 a^2 b x^4-a^3 x^6}{5 b^4 \sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {4 b^3-3 a b^2 x^2+2 a^2 b x^4-a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {b-a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {4 b^3}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}-\frac {3 a b^2 x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}+\frac {2 a^2 b x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}-\frac {a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{b+2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {4 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.18, size = 574, normalized size = 2.40 \begin {gather*} \frac {2 \sqrt [10]{-1} x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (\left (1+\sqrt [5]{-1}\right )^2 \left (1-3 \sqrt [5]{-1}+(-1)^{2/5}\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+2 \left ((-1)^{2/5} \Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\left (1-\sqrt [5]{-1}+2 (-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) \Pi \left (-\sqrt [10]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-(-1)^{4/5} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{3/5} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{2/5} \Pi \left ((-1)^{7/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-(-1)^{4/5} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{3/5} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )\right )}{\left (\sqrt [5]{-1}-1\right )^2 \left (1+\sqrt [5]{-1}\right )^4 \left (1+(-1)^{2/5}\right ) \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \sqrt {\frac {i b}{a}} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.26, size = 239, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.00, size = 2091, normalized size = 8.75
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{5} x^{5} - b^{5}}{{\left (a^{5} x^{5} + b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 487, normalized size = 2.04
method | result | size |
default | \(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}-\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}-a^{3} b \,\textit {\_Z}^{3}+a^{2} b^{2} \textit {\_Z}^{2}-a \,b^{3} \textit {\_Z} +b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+4 b^{3}\right ) \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a b +i b^{2}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha \left (i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a b -b^{2}\right ) a}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{\left (4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-b^{3}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{5 b^{2}}-\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}+\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{5 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}+\frac {b}{a}\right )}\) | \(487\) |
elliptic | \(\text {Expression too large to display}\) | \(2185\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{5} x^{5} - b^{5}}{{\left (a^{5} x^{5} + b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - b\right ) \left (a^{4} x^{4} + a^{3} b x^{3} + a^{2} b^{2} x^{2} + a b^{3} x + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right ) \left (a^{4} x^{4} - a^{3} b x^{3} + a^{2} b^{2} x^{2} - a b^{3} x + b^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________