Optimal. Leaf size=216 \[ -\frac {2 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt {x} (a+b x)}+\frac {6 a b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {2 b^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 216, 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 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt {x} (a+b x)}+\frac {6 a b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {2 b^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}-\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (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}}{x^{5/2}} \, 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)}{x^{5/2}} \, 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}{x^{5/2}}+\frac {a^2 b^3 (3 A b+a B)}{x^{3/2}}+\frac {3 a b^4 (A b+a B)}{\sqrt {x}}+b^5 (A b+3 a B) \sqrt {x}+b^6 B x^{3/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {6 a b (A b+a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^2 (A b+3 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 84, normalized size = 0.39 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (5 a^3 (A+3 B x)+45 a^2 b x (A-B x)-15 a b^2 x^2 (3 A+B x)-b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.93, size = 97, normalized size = 0.45 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-5 a^3 A-15 a^3 B x-45 a^2 A b x+45 a^2 b B x^2+45 a A b^2 x^2+15 a b^2 B x^3+5 A b^3 x^3+3 b^3 B x^4\right )}{15 x^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 73, normalized size = 0.34 \begin {gather*} \frac {2 \, {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 123, normalized size = 0.57 \begin {gather*} \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a b^{2} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (3 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 9 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + A a^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac {3}{2}}} \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 = 0.43 \begin {gather*} -\frac {2 \left (-3 B \,b^{3} x^{4}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 A a \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 A \,a^{2} b x +15 B \,a^{3} x +5 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15 \left (b x +a \right )^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 130, normalized size = 0.60 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, b^{3} x^{2} + 5 \, a b^{2} x\right )} \sqrt {x} + \frac {10 \, {\left (a b^{2} x^{2} + 3 \, a^{2} b x\right )}}{\sqrt {x}} + \frac {15 \, {\left (a^{2} b x^{2} - a^{3} x\right )}}{x^{\frac {3}{2}}}\right )} B + \frac {2}{3} \, A {\left (\frac {b^{3} x^{2} + 3 \, a b^{2} x}{\sqrt {x}} + \frac {6 \, {\left (a b^{2} x^{2} - a^{2} b x\right )}}{x^{\frac {3}{2}}} - \frac {3 \, a^{2} b x^{2} + a^{3} x}{x^{\frac {5}{2}}}\right )} \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}}{x^{5/2}} \,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 \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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