Optimal. Leaf size=123 \[ -\frac {a^3 x (a+b x)^{n+1}}{b^4 (n+1) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{n+2}}{b^4 (n+2) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{n+3}}{b^4 (n+3) \sqrt {c x^2}}+\frac {x (a+b x)^{n+4}}{b^4 (n+4) \sqrt {c x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^3 x (a+b x)^{n+1}}{b^4 (n+1) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{n+2}}{b^4 (n+2) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{n+3}}{b^4 (n+3) \sqrt {c x^2}}+\frac {x (a+b x)^{n+4}}{b^4 (n+4) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx &=\frac {x \int x^3 (a+b x)^n \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a^3 (a+b x)^n}{b^3}+\frac {3 a^2 (a+b x)^{1+n}}{b^3}-\frac {3 a (a+b x)^{2+n}}{b^3}+\frac {(a+b x)^{3+n}}{b^3}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a^3 x (a+b x)^{1+n}}{b^4 (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 (4+n) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 96, normalized size = 0.78 \[ \frac {x (a+b x)^{n+1} \left (-6 a^3+6 a^2 b (n+1) x-3 a b^2 \left (n^2+3 n+2\right ) x^2+b^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )}{b^4 (n+1) (n+2) (n+3) (n+4) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 158, normalized size = 1.28 \[ \frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{4} c n^{4} + 10 \, b^{4} c n^{3} + 35 \, b^{4} c n^{2} + 50 \, b^{4} c n + 24 \, b^{4} c\right )} x} \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{4}}{\sqrt {c x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.09 \[ -\frac {\left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right ) x \left (b x +a \right )^{n +1}}{\sqrt {c \,x^{2}}\, \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 104, normalized size = 0.85 \[ \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 186, normalized size = 1.51 \[ \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^5\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4\,x}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x^2}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^4\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^3\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{\sqrt {c\,x^2}} \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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