Optimal. Leaf size=878 \[ -\frac {b \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{8 (-a)^{7/4} \sqrt {b c-a d}}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{8 (-a)^{7/4} \sqrt {a d-b c}}-\frac {d^{3/4} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{12 a c^{5/4} \sqrt {d x^8+c}}-\frac {b \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {b \left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {b \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}-\frac {\sqrt {d x^8+c}}{6 a c x^6} \]
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Rubi [A] time = 1.22, antiderivative size = 878, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {465, 480, 523, 220, 409, 1217, 1707} \[ -\frac {b \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{8 (-a)^{7/4} \sqrt {b c-a d}}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{8 (-a)^{7/4} \sqrt {a d-b c}}-\frac {d^{3/4} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{12 a c^{5/4} \sqrt {d x^8+c}}-\frac {b \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {b \left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {b \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}-\frac {\sqrt {d x^8+c}}{6 a c x^6} \]
Antiderivative was successfully verified.
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Rule 220
Rule 409
Rule 465
Rule 480
Rule 523
Rule 1217
Rule 1707
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^8}}{6 a c x^6}+\frac {\operatorname {Subst}\left (\int \frac {-3 b c-a d-b d x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {\sqrt {c+d x^8}}{6 a c x^6}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{2 a}-\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {\sqrt {c+d x^8}}{6 a c x^6}-\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{12 a c^{5/4} \sqrt {c+d x^8}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac {\sqrt {c+d x^8}}{6 a c x^6}-\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{12 a c^{5/4} \sqrt {c+d x^8}}-\frac {\left (b^{3/2} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a^2 (b c+a d)}-\frac {\left (b^{3/2} \sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a^2 (b c+a d)}-\frac {\left (b \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a (b c+a d)}-\frac {\left (b \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,x^2\right )}{4 a^2 (b c+a d)}\\ &=-\frac {\sqrt {c+d x^8}}{6 a c x^6}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{7/4} \sqrt {b c-a d}}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{7/4} \sqrt {-b c+a d}}-\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{12 a c^{5/4} \sqrt {c+d x^8}}-\frac {b \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}-\frac {b \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a^2 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}-\frac {b \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}}-\frac {b \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 141, normalized size = 0.16 \[ \frac {-5 x^8 \sqrt {\frac {d x^8}{c}+1} (a d+3 b c) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-b d x^{16} \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-5 a \left (c+d x^8\right )}{30 a^2 c x^6 \sqrt {c+d x^8}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}\, x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^7\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{7} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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