Optimal. Leaf size=65 \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]
[Out]
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Rubi [A] time = 0.0668694, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 20.0883, size = 53, normalized size = 0.82 \[ \frac{c \sqrt{c x^{2}} \left (d x\right )^{m + 4} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 4 \\ m + 5 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{d^{4} x \left (m + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a)**n,x)
[Out]
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Mathematica [A] time = 0.0574414, size = 57, normalized size = 0.88 \[ \frac{x \left (c x^2\right )^{3/2} (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{m+4} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^n,x]
[Out]
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Maple [F] time = 0.052, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m} c x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="giac")
[Out]