Optimal. Leaf size=97 \[ \frac {a^2 c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {2 a b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac {b^2 c \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]
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Rubi [A] time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 43} \begin {gather*} \frac {a^2 c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {2 a b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac {b^2 c \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) 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)^2 \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x)^2 \, dx}{d^3 x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 (d x)^{3+m}+\frac {2 a b (d x)^{4+m}}{d}+\frac {b^2 (d x)^{5+m}}{d^2}\right ) \, dx}{d^3 x}\\ &=\frac {a^2 c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {2 a b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x}+\frac {b^2 c (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.49 \begin {gather*} x \left (c x^2\right )^{3/2} (d x)^m \left (\frac {a^2}{m+4}+\frac {2 a b x}{m+5}+\frac {b^2 x^2}{m+6}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.42, size = 105, normalized size = 1.08 \begin {gather*} \frac {{\left ({\left (b^{2} c m^{2} + 9 \, b^{2} c m + 20 \, b^{2} c\right )} x^{5} + 2 \, {\left (a b c m^{2} + 10 \, a b c m + 24 \, a b c\right )} x^{4} + {\left (a^{2} c m^{2} + 11 \, a^{2} c m + 30 \, a^{2} c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 15 \, m^{2} + 74 \, m + 120} \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 = 95, normalized size = 0.98 \begin {gather*} \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +9 b^{2} m \,x^{2}+a^{2} m^{2}+20 a b m x +20 b^{2} x^{2}+11 a^{2} m +48 a b x +30 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x \left (d x \right )^{m}}{\left (m +6\right ) \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 = 64, normalized size = 0.66 \begin {gather*} \frac {b^{2} c^{\frac {3}{2}} d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, a b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a^{2} 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 = 0.28, size = 121, normalized size = 1.25 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {a^2\,c\,x^3\,\sqrt {c\,x^2}\,\left (m^2+11\,m+30\right )}{m^3+15\,m^2+74\,m+120}+\frac {b^2\,c\,x^5\,\sqrt {c\,x^2}\,\left (m^2+9\,m+20\right )}{m^3+15\,m^2+74\,m+120}+\frac {2\,a\,b\,c\,x^4\,\sqrt {c\,x^2}\,\left (m^2+10\,m+24\right )}{m^3+15\,m^2+74\,m+120}\right ) \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^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx}{d^{6}} & \text {for}\: m = -6 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{d^{5}} & \text {for}\: m = -5 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {a^{2} c^{\frac {3}{2}} d^{m} m^{2} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {11 a^{2} c^{\frac {3}{2}} d^{m} m x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {30 a^{2} c^{\frac {3}{2}} d^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {2 a b c^{\frac {3}{2}} d^{m} m^{2} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 a b c^{\frac {3}{2}} d^{m} m x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {48 a b c^{\frac {3}{2}} d^{m} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {b^{2} c^{\frac {3}{2}} d^{m} m^{2} x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {9 b^{2} c^{\frac {3}{2}} d^{m} m x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 b^{2} c^{\frac {3}{2}} d^{m} x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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