Optimal. Leaf size=103 \[ \frac {a^2 c^2 \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac {2 a b c^2 \sqrt {c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac {b^2 c^2 \sqrt {c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]
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Rubi [A] time = 0.05, antiderivative size = 103, 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^2 \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac {2 a b c^2 \sqrt {c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac {b^2 c^2 \sqrt {c x^2} (d x)^{m+8}}{d^8 (m+8) 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 )^{5/2} (a+b x)^2 \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int (d x)^{5+m} (a+b x)^2 \, dx}{d^5 x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 (d x)^{5+m}+\frac {2 a b (d x)^{6+m}}{d}+\frac {b^2 (d x)^{7+m}}{d^2}\right ) \, dx}{d^5 x}\\ &=\frac {a^2 c^2 (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}+\frac {2 a b c^2 (d x)^{7+m} \sqrt {c x^2}}{d^7 (7+m) x}+\frac {b^2 c^2 (d x)^{8+m} \sqrt {c x^2}}{d^8 (8+m) x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.47 \begin {gather*} x \left (c x^2\right )^{5/2} (d x)^m \left (\frac {a^2}{m+6}+\frac {2 a b x}{m+7}+\frac {b^2 x^2}{m+8}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.28, size = 123, normalized size = 1.19 \begin {gather*} \frac {{\left ({\left (b^{2} c^{2} m^{2} + 13 \, b^{2} c^{2} m + 42 \, b^{2} c^{2}\right )} x^{7} + 2 \, {\left (a b c^{2} m^{2} + 14 \, a b c^{2} m + 48 \, a b c^{2}\right )} x^{6} + {\left (a^{2} c^{2} m^{2} + 15 \, a^{2} c^{2} m + 56 \, a^{2} c^{2}\right )} x^{5}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 21 \, m^{2} + 146 \, m + 336} \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 = 0.92 \begin {gather*} \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +13 b^{2} m \,x^{2}+a^{2} m^{2}+28 a b m x +42 b^{2} x^{2}+15 a^{2} m +96 a b x +56 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}} x \left (d x \right )^{m}}{\left (m +8\right ) \left (m +7\right ) \left (m +6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 64, normalized size = 0.62 \begin {gather*} \frac {b^{2} c^{\frac {5}{2}} d^{m} x^{8} x^{m}}{m + 8} + \frac {2 \, a b c^{\frac {5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac {a^{2} c^{\frac {5}{2}} d^{m} x^{6} x^{m}}{m + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 127, normalized size = 1.23 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {a^2\,c^2\,x^5\,\sqrt {c\,x^2}\,\left (m^2+15\,m+56\right )}{m^3+21\,m^2+146\,m+336}+\frac {b^2\,c^2\,x^7\,\sqrt {c\,x^2}\,\left (m^2+13\,m+42\right )}{m^3+21\,m^2+146\,m+336}+\frac {2\,a\,b\,c^2\,x^6\,\sqrt {c\,x^2}\,\left (m^2+14\,m+48\right )}{m^3+21\,m^2+146\,m+336}\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 {5}{2}}}{x^{8}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx}{d^{8}} & \text {for}\: m = -8 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{5}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx}{d^{7}} & \text {for}\: m = -7 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{5}}\, dx}{d^{6}} & \text {for}\: m = -6 \\\frac {a^{2} c^{\frac {5}{2}} d^{m} m^{2} x x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {15 a^{2} c^{\frac {5}{2}} d^{m} m x x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {56 a^{2} c^{\frac {5}{2}} d^{m} x x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {2 a b c^{\frac {5}{2}} d^{m} m^{2} x^{2} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {28 a b c^{\frac {5}{2}} d^{m} m x^{2} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {96 a b c^{\frac {5}{2}} d^{m} x^{2} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {b^{2} c^{\frac {5}{2}} d^{m} m^{2} x^{3} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {13 b^{2} c^{\frac {5}{2}} d^{m} m x^{3} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {42 b^{2} c^{\frac {5}{2}} d^{m} x^{3} x^{m} \left (x^{2}\right )^{\frac {5}{2}}}{m^{3} + 21 m^{2} + 146 m + 336} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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