Optimal. Leaf size=95 \[ \frac {2 a \sqrt {c+d x^n}}{3 b n (b c-a d) \left (a+b x^n\right )^{3/2}}-\frac {2 (3 b c-a d) \sqrt {c+d x^n}}{3 b n (b c-a d)^2 \sqrt {a+b x^n}} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 78, 37} \begin {gather*} \frac {2 a \sqrt {c+d x^n}}{3 b n (b c-a d) \left (a+b x^n\right )^{3/2}}-\frac {2 (3 b c-a d) \sqrt {c+d x^n}}{3 b n (b c-a d)^2 \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
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 {2 a \sqrt {c+d x^n}}{3 b (b c-a d) n \left (a+b x^n\right )^{3/2}}+\frac {(3 b c-a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^n\right )}{3 b (b c-a d) n}\\ &=\frac {2 a \sqrt {c+d x^n}}{3 b (b c-a d) n \left (a+b x^n\right )^{3/2}}-\frac {2 (3 b c-a d) \sqrt {c+d x^n}}{3 b (b c-a d)^2 n \sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 57, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {c+d x^n} \left (-2 a c+a d x^n-3 b c x^n\right )}{3 n (b c-a d)^2 \left (a+b x^n\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.86, 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 = 0.74, size = 135, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (2 \, a c + {\left (3 \, b c - a d\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{3 \, {\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} n x^{2 \, n} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} n x^{n} + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} 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 {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \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 = 0.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2 n -1}}{\left (b \,x^{n}+a \right )^{\frac {5}{2}} \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 {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \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 )}^{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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