Optimal. Leaf size=210 \[ \frac {b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac {3 a b x^7 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {a^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac {b^3 B x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a^3 A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} \frac {b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac {3 a b x^7 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {a^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac {a^3 A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {b^3 B x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^4 (A+B x) \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 x^4 \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^3 A b^3 x^4+a^2 b^3 (3 A b+a B) x^5+3 a b^4 (A b+a B) x^6+b^5 (A b+3 a B) x^7+b^6 B x^8\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {a^3 A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {a^2 (3 A b+a B) x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {3 a b (A b+a B) x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {b^2 (A b+3 a B) x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {b^3 B x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.41 \begin {gather*} \frac {x^5 \sqrt {(a+b x)^2} \left (84 a^3 (6 A+5 B x)+180 a^2 b x (7 A+6 B x)+135 a b^2 x^2 (8 A+7 B x)+35 b^3 x^3 (9 A+8 B x)\right )}{2520 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.03, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 73, normalized size = 0.35 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} + \frac {1}{5} \, A a^{3} x^{5} + \frac {1}{8} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + \frac {3}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 150, normalized size = 0.71 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, B a b^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B a^{2} b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, A a b^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{2} b x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (5 \, B a^{9} - 9 \, A a^{8} b\right )} \mathrm {sgn}\left (b x + a\right )}{2520 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.44 \begin {gather*} \frac {\left (280 b^{3} B \,x^{4}+315 A \,b^{3} x^{3}+945 x^{3} B a \,b^{2}+1080 x^{2} A a \,b^{2}+1080 B \,a^{2} b \,x^{2}+1260 x A \,a^{2} b +420 B \,a^{3} x +504 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x^{5}}{2520 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 361, normalized size = 1.72 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B x^{4}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a x^{3}}{72 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A x^{3}}{8 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{5} x}{4 \, b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{4} x}{4 \, b^{4}} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} x^{2}}{168 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a x^{2}}{56 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{6}}{4 \, b^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{5}}{4 \, b^{5}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} x}{504 \, b^{5}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{2} x}{56 \, b^{4}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4}}{504 \, b^{6}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{3}}{280 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \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*} \int x^{4} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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