Optimal. Leaf size=218 \[ \frac {2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac {6 a b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {2 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 (a+b x)}+\frac {2 b^3 B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 a^3 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.08, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} \frac {2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac {6 a b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {2 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 (a+b x)}+\frac {2 a^3 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^3 B x^{9/2} \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 \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{\sqrt {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 (\frac {a^3 A b^3}{\sqrt {x}}+a^2 b^3 (3 A b+a B) \sqrt {x}+3 a b^4 (A b+a B) x^{3/2}+b^5 (A b+3 a B) x^{5/2}+b^6 B x^{7/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 a^3 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 a^2 (3 A b+a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {6 a b (A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b^2 (A b+3 a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b^3 B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 88, normalized size = 0.40 \begin {gather*} \frac {2 \sqrt {x} \sqrt {(a+b x)^2} \left (105 a^3 (3 A+B x)+63 a^2 b x (5 A+3 B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right )}{315 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 7.34, size = 115, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (315 a^3 A \sqrt {x}+105 a^3 B x^{3/2}+315 a^2 A b x^{3/2}+189 a^2 b B x^{5/2}+189 a A b^2 x^{5/2}+135 a b^2 B x^{7/2}+45 A b^3 x^{7/2}+35 b^3 B x^{9/2}\right )}{315 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 73, normalized size = 0.33 \begin {gather*} \frac {2}{315} \, {\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 125, normalized size = 0.57 \begin {gather*} \frac {2}{9} \, B b^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, B a b^{2} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{2} b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) \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 = 0.42 \begin {gather*} \frac {2 \left (35 B \,b^{3} x^{4}+45 A \,b^{3} x^{3}+135 B a \,b^{2} x^{3}+189 A a \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+315 A \,a^{2} b x +105 B \,a^{3} x +315 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \sqrt {x}}{315 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 136, normalized size = 0.62 \begin {gather*} \frac {2}{105} \, {\left (3 \, {\left (5 \, b^{3} x^{2} + 7 \, a b^{2} x\right )} x^{\frac {3}{2}} + 14 \, {\left (3 \, a b^{2} x^{2} + 5 \, a^{2} b x\right )} \sqrt {x} + \frac {35 \, {\left (a^{2} b x^{2} + 3 \, a^{3} x\right )}}{\sqrt {x}}\right )} A + \frac {2}{315} \, {\left (5 \, {\left (7 \, b^{3} x^{2} + 9 \, a b^{2} x\right )} x^{\frac {5}{2}} + 18 \, {\left (5 \, a b^{2} x^{2} + 7 \, a^{2} b x\right )} x^{\frac {3}{2}} + 21 \, {\left (3 \, a^{2} b x^{2} + 5 \, a^{3} x\right )} \sqrt {x}\right )} B \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 \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{\sqrt {x}} \,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 \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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