Optimal. Leaf size=149 \[ -\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^8} (3 b c-2 a d)}{8 a^2 c x^4 (b c-a d)}+\frac {b \sqrt {c+d x^8}}{8 a x^4 \left (a+b x^8\right ) (b c-a d)} \]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 149, 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, 472, 583, 12, 377, 205} \begin {gather*} -\frac {\sqrt {c+d x^8} (3 b c-2 a d)}{8 a^2 c x^4 (b c-a d)}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac {b \sqrt {c+d x^8}}{8 a x^4 \left (a+b x^8\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 377
Rule 465
Rule 472
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^4\right )\\ &=\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) x^4 \left (a+b x^8\right )}-\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^4\right )}{8 a (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) x^4}+\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) x^4 \left (a+b x^8\right )}-\frac {\operatorname {Subst}\left (\int \frac {b c (3 b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{8 a^2 c (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) x^4}+\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) x^4 \left (a+b x^8\right )}-\frac {(b (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{8 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) x^4}+\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) x^4 \left (a+b x^8\right )}-\frac {(b (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^4}{\sqrt {c+d x^8}}\right )}{8 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) x^4}+\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) x^4 \left (a+b x^8\right )}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.03, size = 869, normalized size = 5.83 \begin {gather*} -\frac {\sqrt {d x^8+c} \left (120 d^2 \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}}\right ) x^{16}+96 d^2 \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right ) x^{16}+32 d^2 \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right ) x^{16}-120 d^2 \sqrt {\frac {a (b c-a d) x^8 \left (d x^8+c\right )}{c^2 \left (b x^8+a\right )^2}} x^{16}+180 c d \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}}\right ) x^8+160 c d \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right ) x^8+64 c d \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right ) x^8-180 c d \sqrt {\frac {a (b c-a d) x^8 \left (d x^8+c\right )}{c^2 \left (b x^8+a\right )^2}} x^8+45 c^2 \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}}\right )+64 c^2 \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )+32 c^2 \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )-45 c^2 \sqrt {\frac {a (b c-a d) x^8 \left (d x^8+c\right )}{c^2 \left (b x^8+a\right )^2}}\right )}{120 c^3 x^4 \left (\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )^{3/2} \left (b x^8+a\right )^2 \sqrt {\frac {a \left (d x^8+c\right )}{c \left (b x^8+a\right )}}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.97, size = 159, normalized size = 1.07 \begin {gather*} \frac {\left (4 a b d-3 b^2 c\right ) \tan ^{-1}\left (\frac {a \sqrt {d}+b x^4 \sqrt {c+d x^8}+b \sqrt {d} x^8}{\sqrt {a} \sqrt {b c-a d}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^8} \left (-2 a^2 d+2 a b c-2 a b d x^8+3 b^2 c x^8\right )}{8 a^2 c x^4 \left (a+b x^8\right ) (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 612, normalized size = 4.11 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{12} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{4}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 4 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{8} + 2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2}\right )} \sqrt {d x^{8} + c}}{32 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{12} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{4}\right )}}, -\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{12} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{4}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{8} + 2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2}\right )} \sqrt {d x^{8} + c}}{16 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{12} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.76, size = 418, normalized size = 2.81 \begin {gather*} \frac {1}{8} \, d^{\frac {5}{2}} {\left (\frac {{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b^{2} c - 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} a b d - 6 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b^{2} c^{2} + 14 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a b c d - 8 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a^{2} d^{2} + 3 \, b^{2} c^{3} - 2 \, a b c^{2} d\right )}}{{\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a c d - b c^{3}\right )} {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}\, x^{5}}\, dx \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^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{5}}\,{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^5\,{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________