Optimal. Leaf size=61 \[ \frac {2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {794, 648} \begin {gather*} \frac {2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx &=\frac {2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}+\frac {\left (2 \left (-\frac {3}{2} (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{7 c}\\ &=-\frac {2 (2 b B-7 A c) \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.61 \begin {gather*} \frac {2 (x (b+c x))^{5/2} (7 A c-2 b B+5 B c x)}{35 c^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.51, size = 39, normalized size = 0.64 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{5/2} (7 A c-2 b B+5 B c x)}{35 c^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 76, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (5 \, B c^{3} x^{3} - 2 \, B b^{3} + 7 \, A b^{2} c + {\left (8 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} + {\left (B b^{2} c + 14 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{35 \, c^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 149, normalized size = 2.44 \begin {gather*} -\frac {2}{105} \, B c {\left (\frac {8 \, b^{\frac {7}{2}}}{c^{3}} - \frac {15 \, {\left (c x + b\right )}^{\frac {7}{2}} - 42 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}}{c^{3}}\right )} + \frac {2}{15} \, B b {\left (\frac {2 \, b^{\frac {5}{2}}}{c^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b}{c^{2}}\right )} + \frac {2}{15} \, A c {\left (\frac {2 \, b^{\frac {5}{2}}}{c^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b}{c^{2}}\right )} + \frac {2}{3} \, A b {\left (\frac {{\left (c x + b\right )}^{\frac {3}{2}}}{c} - \frac {b^{\frac {3}{2}}}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.64 \begin {gather*} \frac {2 \left (c x +b \right ) \left (5 B c x +7 A c -2 b B \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{35 c^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 129, normalized size = 2.11 \begin {gather*} \frac {2 \, {\left (5 \, b c x^{2} + 5 \, b^{2} x + {\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} x\right )} \sqrt {c x + b} A}{15 \, c x} + \frac {2 \, {\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \, {\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt {c x + b} B}{105 \, c^{2} x^{2}} \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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{x^{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 \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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