Optimal. Leaf size=252 \[ \frac {5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac {5 (b c-a d)^2 (a d+7 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n} \]
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Rubi [A] time = 0.23, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac {(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {5 (b c-a d)^2 (a d+7 b c) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 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 )^{5/2}}{\sqrt {c+d x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}-\frac {(7 b c+a d) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{8 b d n}\\ &=-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}+\frac {(5 (b c-a d) (7 b c+a d)) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{48 b d^2 n}\\ &=\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}-\frac {\left (5 (b c-a d)^2 (7 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{64 b d^3 n}\\ &=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{128 b d^4 n}\\ &=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left (5 (b c-a d)^3 (7 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 )}{64 b^2 d^4 n}\\ &=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left (5 (b c-a d)^3 (7 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 )}{64 b^2 d^4 n}\\ &=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b d^4 n}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b d^3 n}-\frac {(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b d^2 n}+\frac {\left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d n}+\frac {5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 223, normalized size = 0.88 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (15 a^3 d^3+a^2 b d^2 \left (118 d x^n-191 c\right )+a b^2 d \left (265 c^2-172 c d x^n+136 d^2 x^{2 n}\right )+b^3 \left (-105 c^3+70 c^2 d x^n-56 c d^2 x^{2 n}+48 d^3 x^{3 n}\right )\right )+15 (a d+7 b c) (b c-a d)^{7/2} \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 )}{192 b^2 d^{9/2} n \sqrt {c+d x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+2 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.54, size = 607, normalized size = 2.41 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{768 \, b^{2} d^{5} n}, -\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{384 \, b^{2} d^{5} 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 {5}{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 = 1.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{n}+a \right )^{\frac {5}{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 {5}{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 = -0.00 \begin {gather*} \int \frac {x^{2\,n-1}\,{\left (a+b\,x^n\right )}^{5/2}}{\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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