Optimal. Leaf size=150 \[ -\frac {3 (b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n} \]
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Rubi [A]
time = 0.10, antiderivative size = 150, 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 = {457, 92, 81, 65,
223, 212} \begin {gather*} -\frac {\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}-\frac {3 (a d+b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}+\frac {\text {Subst}\left (\int \frac {-a c-\frac {3}{2} (b c+a d) x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{2 b d n}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{8 b^2 d^2 n}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {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^3 d^2 n}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {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^3 d^2 n}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d n}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 157, normalized size = 1.05 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (-3 b c-3 a d+2 b d x^n\right )+\sqrt {b c-a d} \left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \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^3 d^{5/2} n \sqrt {c+d x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+3 n}}{\sqrt {a +b \,x^{n}}\, \sqrt {c +d \,x^{n}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.32, size = 361, normalized size = 2.41 \begin {gather*} \left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, 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 - 3 \, a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{16 \, b^{3} d^{3} n}, -\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, 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 - 3 \, a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{8 \, b^{3} d^{3} n}\right ] \end {gather*}
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 \begin {gather*} \int \frac {x^{3 n - 1}}{\sqrt {a + b x^{n}} \sqrt {c + d x^{n}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{3\,n-1}}{\sqrt {a+b\,x^n}\,\sqrt {c+d\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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