Optimal. Leaf size=167 \[ -\frac {a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {d^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.19, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1265, 1660, 12,
738, 212} \begin {gather*} \frac {d^2 \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {x^2 \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 738
Rule 1265
Rule 1660
Rubi steps
\begin {align*} \int \frac {x^5}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {Subst}\left (\int -\frac {\left (b^2-4 a c\right ) d^2}{2 \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=-\frac {a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {d^2 \text {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {d^2 \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x^2}{\sqrt {a+b x^2+c x^4}}\right )}{c d^2-b d e+a e^2}\\ &=-\frac {a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {d^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.65, size = 180, normalized size = 1.08 \begin {gather*} \frac {-2 a^2 e+b^2 d x^2-2 a c d x^2+a b \left (d-e x^2\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+b x^2+c x^4}}+\frac {d^2 \sqrt {-c d^2+e (b d-a e)} \tan ^{-1}\left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (c d^2+e (-b d+a e)\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(541\) vs.
\(2(156)=312\).
time = 0.14, size = 542, normalized size = 3.25
method | result | size |
elliptic | \(\frac {2 c \,d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) e \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right )^{2} \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{2 \left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) c \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right )^{2} \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{2 \left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) c \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\) | \(493\) |
default | \(-\frac {b \,x^{2}+2 a}{e \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {d \left (2 c \,x^{2}+b \right )}{e^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {d^{2} \left (\frac {2 c e \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {2 c \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{e^{2}}\) | \(542\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 677 vs.
\(2 (159) = 318\).
time = 0.60, size = 1399, normalized size = 8.38 \begin {gather*} \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{4} + {\left (b^{3} - 4 \, a b c\right )} d^{2} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) - 4 \, {\left (a b c d^{3} + {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} x^{2} - {\left (a^{2} b x^{2} + 2 \, a^{3}\right )} e^{3} + {\left (3 \, a^{2} b d + 2 \, {\left (a b^{2} - a^{2} c\right )} d x^{2}\right )} e^{2} - {\left ({\left (b^{3} - a b c\right )} d^{2} x^{2} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2}\right )} e\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{4} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{4} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x^{2} + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{4} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x^{2} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{4} + {\left (b^{3} - 4 \, a b c\right )} d^{2} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) - 2 \, {\left (a b c d^{3} + {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} x^{2} - {\left (a^{2} b x^{2} + 2 \, a^{3}\right )} e^{3} + {\left (3 \, a^{2} b d + 2 \, {\left (a b^{2} - a^{2} c\right )} d x^{2}\right )} e^{2} - {\left ({\left (b^{3} - a b c\right )} d^{2} x^{2} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2}\right )} e\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{4} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{4} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x^{2} + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{4} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x^{2} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}\right ] \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 {x^{5}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs.
\(2 (159) = 318\).
time = 4.11, size = 458, normalized size = 2.74 \begin {gather*} \frac {d^{2} \arctan \left (-\frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} - \frac {\frac {{\left (b^{2} c d^{3} - 2 \, a c^{2} d^{3} - b^{3} d^{2} e + a b c d^{2} e + 2 \, a b^{2} d e^{2} - 2 \, a^{2} c d e^{2} - a^{2} b e^{3}\right )} x^{2}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {a b c d^{3} - a b^{2} d^{2} e - 2 \, a^{2} c d^{2} e + 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}}{\sqrt {c x^{4} + b x^{2} + a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{\left (e\,x^2+d\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________