Optimal. Leaf size=115 \[ \frac {b^2 \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac {\sqrt {c+d x^4}}{6 a c x^6} \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 480, 583, 12, 377, 205} \begin {gather*} \frac {b^2 \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac {\sqrt {c+d x^4}}{6 a c x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 377
Rule 465
Rule 480
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {\operatorname {Subst}\left (\int \frac {-3 b c-2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {(3 b c+2 a d) \sqrt {c+d x^4}}{6 a^2 c^2 x^2}-\frac {\operatorname {Subst}\left (\int -\frac {3 b^2 c^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{6 a^2 c^2}\\ &=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {(3 b c+2 a d) \sqrt {c+d x^4}}{6 a^2 c^2 x^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {(3 b c+2 a d) \sqrt {c+d x^4}}{6 a^2 c^2 x^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{2 a^2}\\ &=-\frac {\sqrt {c+d x^4}}{6 a c x^6}+\frac {(3 b c+2 a d) \sqrt {c+d x^4}}{6 a^2 c^2 x^2}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2} \sqrt {b c-a d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.55, size = 137, normalized size = 1.19 \begin {gather*} \frac {\frac {b^2 x^2 \sqrt {\frac {d x^4}{c}+1} \sin ^{-1}\left (\frac {\sqrt {x^4 \left (\frac {b}{a}-\frac {d}{c}\right )}}{\sqrt {\frac {b x^4}{a}+1}}\right )}{a^3 \sqrt {\frac {x^4 (b c-a d)}{a c}}}+\frac {\left (c+d x^4\right ) \left (-a c+2 a d x^4+3 b c x^4\right )}{3 a^2 c^2 x^6}}{2 \sqrt {c+d x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.89, size = 163, normalized size = 1.42 \begin {gather*} \frac {\sqrt {c+d x^4} \left (-a c+2 a d x^4+3 b c x^4\right )}{6 a^2 c^2 x^6}-\frac {b^2 \sqrt {b c-a d} \tan ^{-1}\left (\frac {b \sqrt {d} x^4}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^2 \sqrt {c+d x^4}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{2 a^{5/2} (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 418, normalized size = 3.63 \begin {gather*} \left [-\frac {3 \, \sqrt {-a b c + a^{2} d} b^{2} c^{2} x^{6} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}, \frac {3 \, \sqrt {a b c - a^{2} d} b^{2} c^{2} x^{6} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) - 2 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.68, size = 205, normalized size = 1.78 \begin {gather*} -\frac {1}{6} \, d^{\frac {5}{2}} {\left (\frac {3 \, b^{2} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.27, size = 383, normalized size = 3.33 \begin {gather*} -\frac {b^{2} \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {b^{2} \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {\sqrt {d \,x^{4}+c}\, b}{2 a^{2} c \,x^{2}}-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 d \,x^{4}+c \right )}{6 a \,c^{2} x^{6}} \end {gather*}
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 {1}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c} x^{7}}\,{d x} \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 {1}{x^7\,\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \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 {1}{x^{7} \left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________