Optimal. Leaf size=112 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (A b-a B) (a c+b c x)^{m+4}}{b^2 c^4 (m+4) (a+b x)}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (a c+b c x)^{m+5}}{b^2 c^5 (m+5) (a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {770, 21, 43} \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (A b-a B) (a c+b c x)^{m+4}}{b^2 c^4 (m+4) (a+b x)}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (a c+b c x)^{m+5}}{b^2 c^5 (m+5) (a+b x)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (A+B x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (a c+b c x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (A+B x) (a c+b c x)^{3+m} \, dx}{c^3 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(A b-a B) (a c+b c x)^{3+m}}{b}+\frac {B (a c+b c x)^{4+m}}{b c}\right ) \, dx}{c^3 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a c+b c x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{b^2 c^4 (4+m) (a+b x)}+\frac {B (a c+b c x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{b^2 c^5 (5+m) (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.53 \[ \frac {(a+b x)^3 \sqrt {(a+b x)^2} (c (a+b x))^m (-a B+A b (m+5)+b B (m+4) x)}{b^2 (m+4) (m+5)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 219, normalized size = 1.96 \[ \frac {{\left (A a^{4} b m - B a^{5} + 5 \, A a^{4} b + {\left (B b^{5} m + 4 \, B b^{5}\right )} x^{5} + {\left (15 \, B a b^{4} + 5 \, A b^{5} + {\left (4 \, B a b^{4} + A b^{5}\right )} m\right )} x^{4} + 2 \, {\left (10 \, B a^{2} b^{3} + 10 \, A a b^{4} + {\left (3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} m\right )} x^{3} + 2 \, {\left (5 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} + {\left (2 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} m\right )} x^{2} + {\left (20 \, A a^{3} b^{2} + {\left (B a^{4} b + 4 \, A a^{3} b^{2}\right )} m\right )} x\right )} {\left (b c x + a c\right )}^{m}}{b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 545, normalized size = 4.87 \[ \frac {{\left (b c x + a c\right )}^{m} B b^{5} m x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (b c x + a c\right )}^{m} B a b^{4} m x^{4} \mathrm {sgn}\left (b x + a\right ) + {\left (b c x + a c\right )}^{m} A b^{5} m x^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (b c x + a c\right )}^{m} B b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (b c x + a c\right )}^{m} B a^{2} b^{3} m x^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (b c x + a c\right )}^{m} A a b^{4} m x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, {\left (b c x + a c\right )}^{m} B a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (b c x + a c\right )}^{m} A b^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (b c x + a c\right )}^{m} B a^{3} b^{2} m x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (b c x + a c\right )}^{m} A a^{2} b^{3} m x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, {\left (b c x + a c\right )}^{m} B a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, {\left (b c x + a c\right )}^{m} A a b^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (b c x + a c\right )}^{m} B a^{4} b m x \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (b c x + a c\right )}^{m} A a^{3} b^{2} m x \mathrm {sgn}\left (b x + a\right ) + 10 \, {\left (b c x + a c\right )}^{m} B a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, {\left (b c x + a c\right )}^{m} A a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (b c x + a c\right )}^{m} A a^{4} b m \mathrm {sgn}\left (b x + a\right ) + 20 \, {\left (b c x + a c\right )}^{m} A a^{3} b^{2} x \mathrm {sgn}\left (b x + a\right ) - {\left (b c x + a c\right )}^{m} B a^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (b c x + a c\right )}^{m} A a^{4} b \mathrm {sgn}\left (b x + a\right )}{b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 0.55 \[ \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (B b m x +A b m +4 B b x +5 A b -B a \right ) \left (b x +a \right ) \left (b c x +a c \right )^{m}}{\left (m^{2}+9 m +20\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 180, normalized size = 1.61 \[ \frac {{\left (b^{4} c^{m} x^{4} + 4 \, a b^{3} c^{m} x^{3} + 6 \, a^{2} b^{2} c^{m} x^{2} + 4 \, a^{3} b c^{m} x + a^{4} c^{m}\right )} {\left (b x + a\right )}^{m} A}{b {\left (m + 4\right )}} + \frac {{\left (b^{5} c^{m} {\left (m + 4\right )} x^{5} + a b^{4} c^{m} {\left (4 \, m + 15\right )} x^{4} + 2 \, a^{2} b^{3} c^{m} {\left (3 \, m + 10\right )} x^{3} + 2 \, a^{3} b^{2} c^{m} {\left (2 \, m + 5\right )} x^{2} + a^{4} b c^{m} m x - a^{5} c^{m}\right )} {\left (b x + a\right )}^{m} B}{{\left (m^{2} + 9 \, m + 20\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.30, size = 254, normalized size = 2.27 \[ {\left (a\,c+b\,c\,x\right )}^m\,\left (\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (5\,A\,b-B\,a+A\,b\,m\right )}{b^2\,\left (m^2+9\,m+20\right )}+\frac {3\,a\,x^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (5\,A\,b+3\,B\,a+A\,b\,m+B\,a\,m\right )}{m^2+9\,m+20}+\frac {b\,x^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (5\,A\,b+11\,B\,a+A\,b\,m+3\,B\,a\,m\right )}{m^2+9\,m+20}+\frac {a^2\,x\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (15\,A\,b+B\,a+3\,A\,b\,m+B\,a\,m\right )}{b\,\left (m^2+9\,m+20\right )}+\frac {B\,b^2\,x^4\,\left (m+4\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{m^2+9\,m+20}\right ) \]
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 \[ \int \left (c \left (a + b x\right )\right )^{m} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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