Optimal. Leaf size=199 \[ -\frac {(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}+\frac {(b c-a d) (a d+5 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}-\frac {(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n} \]
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Rubi [A] time = 0.16, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}-\frac {(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {(b c-a d) (a d+5 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n} \end {gather*}
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} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}-\frac {(5 b c+a d) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{6 b d n}\\ &=-\frac {(5 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}+\frac {((b c-a d) (5 b c+a d)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{8 b d^2 n}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}-\frac {(5 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{16 b d^3 n}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}-\frac {(5 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \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 )}{8 b^2 d^3 n}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}-\frac {(5 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}-\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \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 )}{8 b^2 d^3 n}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{8 b d^3 n}-\frac {(5 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{12 b d^2 n}+\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d n}-\frac {(b c-a d)^2 (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 178, normalized size = 0.89 \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 (7 d x^n-11 c\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)^{5/2} (a d+5 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 )}{24 b^2 d^{7/2} n \sqrt {c+d x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+2 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.50, size = 469, normalized size = 2.36 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, 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 - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{96 \, b^{2} d^{4} n}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, 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 - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{48 \, b^{2} d^{4} n}\right ] \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*} \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{2 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{n}+a \right )^{\frac {3}{2}} x^{2 n -1}}{\sqrt {d \,x^{n}+c}}\, dx \end {gather*}
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*} \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{2 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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