Optimal. Leaf size=89 \[ \frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} d^{3/2} n} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 80, 63, 217, 206} \[ \frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} d^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{2 b d n}\\ &=\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^n}\right )}{b^2 d n}\\ &=\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^n}}{\sqrt {c+d x^n}}\right )}{b^2 d n}\\ &=\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} d^{3/2} n}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 123, normalized size = 1.38 \[ \frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right )-\sqrt {b c-a d} (a d+b c) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{b^2 d^{3/2} n \sqrt {c+d x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 281, normalized size = 3.16 \[ \left [\frac {4 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}{4 \, b^{2} d^{2} n}, \frac {2 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} b d + {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right )}{2 \, b^{2} d^{2} n}\right ] \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.97, size = 0, normalized size = 0.00 \[ \int \frac {x^{2 n -1}}{\sqrt {b \,x^{n}+a}\, \sqrt {d \,x^{n}+c}}\, dx \]
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 \[ \int \frac {x^{2 \, n - 1}}{\sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{2\,n-1}}{\sqrt {a+b\,x^n}\,\sqrt {c+d\,x^n}} \,d x \]
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 \[ \int \frac {x^{2 n - 1}}{\sqrt {a + b x^{n}} \sqrt {c + d x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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