Optimal. Leaf size=221 \[ \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n} \]
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Rubi [A]
time = 0.15, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {457, 92, 81, 52,
65, 223, 212} \begin {gather*} \frac {\left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n}-\frac {(3 a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
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 \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x} \left (-a c-\frac {1}{2} (5 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^n\right )}{3 b d n}\\ &=-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}+\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{8 b^2 d^2 n}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{16 b^2 d^3 n}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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 )}{8 b^3 d^3 n}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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 )}{8 b^3 d^3 n}\\ &=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b^2 d^3 n}-\frac {(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{3 b d n}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 191, normalized size = 0.86 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (-3 a^2 d^2+2 a b d \left (-2 c+d x^n\right )+b^2 \left (15 c^2-10 c d x^n+8 d^2 x^{2 n}\right )\right )-3 (b c-a d)^{3/2} \left (5 b^2 c^2+2 a b c d+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 )}{24 b^3 d^{7/2} 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 = 5.30, size = 471, normalized size = 2.13 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2 \, n} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{96 \, b^{3} d^{4} n}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2 \, n} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{48 \, b^{3} d^{4} 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 = -0.00 \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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