Optimal. Leaf size=36 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt {c}} \]
________________________________________________________________________________________
Rubi [C] time = 1.99, antiderivative size = 289, normalized size of antiderivative = 8.03, number of steps used = 13, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6728, 329, 220, 933, 168, 537} \begin {gather*} -\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c-\sqrt {c^2-4 a b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt {a x^3+b x}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c+\sqrt {c^2-4 a b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt {a x^3+b x}}+\frac {\sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 168
Rule 220
Rule 329
Rule 537
Rule 933
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {-b+a x^2}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {b+a x^2}}-\frac {2 b+c x}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )}\right ) \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {2 b+c x}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )} \, dx}{\sqrt {b x+a x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (\frac {c-\sqrt {-4 a b+c^2}}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}}+\frac {c+\sqrt {-4 a b+c^2}}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}}\right ) \, dx}{\sqrt {b x+a x^3}}+\frac {\left (2 \sqrt {x} \sqrt {b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}+\frac {\left (2 \left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-c+\sqrt {-4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}+\frac {\left (2 \left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-c-\sqrt {-4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c-\sqrt {-4 a b+c^2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt {b x+a x^3}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c+\sqrt {-4 a b+c^2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt {b x+a x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.64, size = 214, normalized size = 5.94 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {\frac {b}{a x^2}+1} \left (-\Pi \left (\frac {2 i \sqrt {a} \sqrt {b}}{\sqrt {c^2-4 a b}-c};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {2 i \sqrt {a} \sqrt {b}}{c+\sqrt {c^2-4 a b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \sqrt {x \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.30, size = 36, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 159, normalized size = 4.42 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{4} - 6 \, a c x^{3} - 6 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2} - 4 \, \sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {-c}}{a^{2} x^{4} + 2 \, a c x^{3} + 2 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {c}}{2 \, {\left (a c x^{3} + b c x\right )}}\right )}{\sqrt {c}}\right ] \end {gather*}
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 x^{2} - b}{\sqrt {a x^{3} + b x} {\left (a x^{2} + c x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 1142, normalized size = 31.72
method | result | size |
elliptic | \(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {-4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {2 \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {-4 a b +c^{2}}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {-4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {-4 a b +c^{2}}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}\) | \(1142\) |
default | \(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {-4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {2 \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {-c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {-4 a b +c^{2}}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {-4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {c +\sqrt {-4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {-4 a b +c^{2}}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {-4 a b +c^{2}}}{2 a}\right )}\) | \(1161\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.25, size = 51, normalized size = 1.42 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}-\frac {c\,x}{2}+\frac {a\,x^2}{2}+\sqrt {c}\,\sqrt {a\,x^3+b\,x}\,1{}\mathrm {i}}{a\,x^2+c\,x+b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \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 {a x^{2} - b}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} + b + c x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________