Optimal. Leaf size=91 \[ \frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} \sqrt {d} n} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, 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 = {457, 79, 65,
223, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} \sqrt {d} n}+\frac {2 a \sqrt {c+d x^n}}{b n (b c-a d) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 212
Rule 223
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{b n}\\ &=\frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {2 \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 )}{b^2 n}\\ &=\frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {2 \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 )}{b^2 n}\\ &=\frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} \sqrt {d} n}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 105, normalized size = 1.15 \begin {gather*} \frac {2 a \sqrt {c+d x^n}}{b (b c-a d) n \sqrt {a+b x^n}}+\frac {\log \left (b c+a d+2 b d x^n+2 \sqrt {b} \sqrt {d} \sqrt {a+b x^n} \sqrt {c+d x^n}\right )}{b^{3/2} \sqrt {d} n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+2 n}}{\left (a +b \,x^{n}\right )^{\frac {3}{2}} \sqrt {c +d \,x^{n}}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs.
\(2 (75) = 150\).
time = 1.09, size = 408, normalized size = 4.48 \begin {gather*} \left [\frac {4 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} a b d + {\left ({\left (b^{2} c - a b d\right )} \sqrt {b d} x^{n} + {\left (a b c - a^{2} d\right )} \sqrt {b d}\right )} \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 )}{2 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} n x^{n} + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} n\right )}}, \frac {2 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} a b d - {\left ({\left (b^{2} c - a b d\right )} \sqrt {-b d} x^{n} + {\left (a b c - a^{2} d\right )} \sqrt {-b d}\right )} \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 )}{{\left (b^{4} c d - a b^{3} d^{2}\right )} n x^{n} + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} n}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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^{2\,n-1}}{{\left (a+b\,x^n\right )}^{3/2}\,\sqrt {c+d\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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