Optimal. Leaf size=146 \[ \frac {(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}-\frac {(a d+3 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n} \]
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Rubi [A] time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {446, 80, 50, 63, 217, 206} \[ \frac {(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}-\frac {(a d+3 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n} \]
Antiderivative was successfully verified.
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Rule 50
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 {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}-\frac {(3 b c+a d) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{4 b d n}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}+\frac {((b c-a d) (3 b c+a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{8 b d^2 n}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}+\frac {((b c-a d) (3 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 )}{4 b^2 d^2 n}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}+\frac {((b c-a d) (3 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 )}{4 b^2 d^2 n}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}+\frac {(b c-a d) (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 141, normalized size = 0.97 \[ \frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (a d-3 b c+2 b d x^n\right )+(a d+3 b c) (b c-a d)^{3/2} \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{4 b^2 d^{5/2} n \sqrt {c+d x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 359, normalized size = 2.46 \[ \left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) - 4 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{16 \, b^{2} d^{3} n}, -\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{8 \, b^{2} d^{3} n}\right ] \]
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 \[ \int \frac {\sqrt {b x^{n} + a} x^{2 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \,x^{n}+a}\, x^{2 n -1}}{\sqrt {d \,x^{n}+c}}\, dx \]
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 \[ \int \frac {\sqrt {b x^{n} + a} x^{2 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \]
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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