Optimal. Leaf size=61 \[ \frac {a c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 43} \begin {gather*} \frac {a c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin {align*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (d x)^m (a+b x) \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x) \, dx}{d^3 x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a (d x)^{3+m}+\frac {b (d x)^{4+m}}{d}\right ) \, dx}{d^3 x}\\ &=\frac {a c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.62 \begin {gather*} \frac {x \left (c x^2\right )^{3/2} (d x)^m (a (m+5)+b (m+4) x)}{(m+4) (m+5)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.49, size = 50, normalized size = 0.82 \begin {gather*} \frac {{\left ({\left (b c m + 4 \, b c\right )} x^{4} + {\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.66 \begin {gather*} \frac {\left (b m x +a m +4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x \left (d x \right )^{m}}{\left (m +5\right ) \left (m +4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 39, normalized size = 0.64 \begin {gather*} \frac {b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a c^{\frac {3}{2}} d^{m} x^{4} x^{m}}{m + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 42, normalized size = 0.69 \begin {gather*} \frac {c\,x^3\,{\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,\left (5\,a+a\,m+4\,b\,x+b\,m\,x\right )}{m^2+9\,m+20} \end {gather*}
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 \begin {gather*} \begin {cases} \frac {\int \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx + \int \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{d^{5}} & \text {for}\: m = -5 \\\frac {\int \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {a c^{\frac {3}{2}} d^{m} m x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {5 a c^{\frac {3}{2}} d^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {b c^{\frac {3}{2}} d^{m} m x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {4 b c^{\frac {3}{2}} d^{m} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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