Optimal. Leaf size=135 \[ -\frac{a^3 c \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]
[Out]
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Rubi [A] time = 0.100177, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^3 c \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]
Antiderivative was successfully verified.
[In] Int[(c*x^2)^(3/2)*(a + b*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 26.2114, size = 119, normalized size = 0.88 \[ - \frac{a^{3} c \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{4} x \left (n + 1\right )} + \frac{3 a^{2} c \sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{4} x \left (n + 2\right )} - \frac{3 a c \sqrt{c x^{2}} \left (a + b x\right )^{n + 3}}{b^{4} x \left (n + 3\right )} + \frac{c \sqrt{c x^{2}} \left (a + b x\right )^{n + 4}}{b^{4} x \left (n + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(3/2)*(b*x+a)**n,x)
[Out]
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Mathematica [A] time = 0.0875822, size = 98, normalized size = 0.73 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (-6 a^3+6 a^2 b (n+1) x-3 a b^2 \left (n^2+3 n+2\right ) x^2+b^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )}{b^4 (n+1) (n+2) (n+3) (n+4) x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^2)^(3/2)*(a + b*x)^n,x]
[Out]
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Maple [A] time = 0.007, size = 136, normalized size = 1. \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{n}^{3}{x}^{3}-6\,{b}^{3}{n}^{2}{x}^{3}+3\,a{b}^{2}{n}^{2}{x}^{2}-11\,{b}^{3}n{x}^{3}+9\,a{b}^{2}n{x}^{2}-6\,{b}^{3}{x}^{3}-6\,{a}^{2}bnx+6\,a{b}^{2}{x}^{2}-6\,{a}^{2}bx+6\,{a}^{3} \right ) }{{x}^{3}{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(3/2)*(b*x+a)^n,x)
[Out]
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Maxima [A] time = 1.40501, size = 157, normalized size = 1.16 \[ \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{\frac{3}{2}} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{\frac{3}{2}} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} c^{\frac{3}{2}} x^{2} + 6 \, a^{3} b c^{\frac{3}{2}} n x - 6 \, a^{4} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217732, size = 221, normalized size = 1.64 \[ \frac{{\left (6 \, a^{3} b c n x - 6 \, a^{4} c +{\left (b^{4} c n^{3} + 6 \, b^{4} c n^{2} + 11 \, b^{4} c n + 6 \, b^{4} c\right )} x^{4} +{\left (a b^{3} c n^{3} + 3 \, a b^{3} c n^{2} + 2 \, a b^{3} c n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} c n^{2} + a^{2} b^{2} c n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(3/2)*(b*x+a)**n,x)
[Out]
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GIAC/XCAS [A] time = 0.210634, size = 437, normalized size = 3.24 \[{\left (\frac{6 \, a^{4} e^{\left (n{\rm ln}\left (a\right )\right )}{\rm sign}\left (x\right )}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} + \frac{b^{4} n^{3} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + a b^{3} n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 6 \, b^{4} n^{2} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 3 \, a b^{3} n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 11 \, b^{4} n x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 3 \, a^{2} b^{2} n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 2 \, a b^{3} n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 6 \, b^{4} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 3 \, a^{2} b^{2} n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 6 \, a^{3} b n x e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 6 \, a^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right )}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}}\right )} c^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n,x, algorithm="giac")
[Out]