Optimal. Leaf size=317 \[ \frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}+\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac {4 (c x)^{-3 n/4}}{3 a c n} \]
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Rubi [A] time = 0.24, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {363, 362, 345, 211, 1165, 628, 1162, 617, 204} \[ \frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}+\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac {4 (c x)^{-3 n/4}}{3 a c n} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 345
Rule 362
Rule 363
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx &=\frac {\left (x^{3 n/4} (c x)^{-3 n/4}\right ) \int \frac {x^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx}{c}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (b x^{3 n/4} (c x)^{-3 n/4}\right ) \int \frac {x^{\frac {1}{4} (-4+n)}}{a+b x^n} \, dx}{a c}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (4 b x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a c n}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (2 b x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}-\frac {\left (2 b x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (\sqrt {b} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}-\frac {\left (\sqrt {b} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}+\frac {\left (b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} a^{7/4} c n}+\frac {\left (b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} a^{7/4} c n}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {\left (\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}+\frac {\left (\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}\\ &=-\frac {4 (c x)^{-3 n/4}}{3 a c n}+\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.12 \[ -\frac {4 x (c x)^{-\frac {3 n}{4}-1} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\frac {b x^n}{a}\right )}{3 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 426, normalized size = 1.34 \[ -\frac {12 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a^{2} b^{2} c^{-2 \, n - \frac {8}{3}} n x^{\frac {1}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (c) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (x)\right )} - a^{2} n x^{\frac {1}{3}} \sqrt {-\frac {a^{3} b^{3} c^{-3 \, n - 4} n^{2} x^{\frac {1}{3}} \sqrt {-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}} - b^{4} c^{-4 \, n - \frac {16}{3}} x e^{\left (-\frac {1}{6} \, {\left (3 \, n + 4\right )} \log \relax (c) - \frac {1}{6} \, {\left (3 \, n + 4\right )} \log \relax (x)\right )}}{x}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}}}{b^{3} c^{-3 \, n - 4}}\right ) - 3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (c) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (x)\right )}}{x}\right ) + 3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} - b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (c) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \relax (x)\right )}}{x}\right ) + 4 \, x e^{\left (-\frac {1}{4} \, {\left (3 \, n + 4\right )} \log \relax (c) - \frac {1}{4} \, {\left (3 \, n + 4\right )} \log \relax (x)\right )}}{3 \, a n} \]
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 \[ \int \frac {\left (c x\right )^{-\frac {3}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {3 n}{4}-1}}{b \,x^{n}+a}\, dx \]
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 \[ -b \int \frac {x^{\frac {1}{4} \, n}}{a b c^{\frac {3}{4} \, n + 1} x x^{n} + a^{2} c^{\frac {3}{4} \, n + 1} x}\,{d x} - \frac {4 \, c^{-\frac {3}{4} \, n - 1}}{3 \, a n x^{\frac {3}{4} \, n}} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.55, size = 309, normalized size = 0.97 \[ \frac {c^{- \frac {3 n}{4}} x^{- \frac {3 n}{4}} \Gamma \left (- \frac {3}{4}\right )}{a c n \Gamma \left (\frac {1}{4}\right )} - \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4}} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} c n \Gamma \left (\frac {1}{4}\right )} + \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4}} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} c n \Gamma \left (\frac {1}{4}\right )} + \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4}} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} c n \Gamma \left (\frac {1}{4}\right )} - \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4}} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} c n \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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