Optimal. Leaf size=35 \[ -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac {A b}{x^{5/2}}+\frac {b B+A c}{x^{3/2}}+\frac {B c}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 A b}{3 x^{3/2}}-\frac {2 (b B+A c)}{\sqrt {x}}+2 B c \sqrt {x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.80 \[ -\frac {2 (A (b+3 c x)+3 B x (b-c x))}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 27, normalized size = 0.77 \[ \frac {2 \, {\left (3 \, B c x^{2} - A b - 3 \, {\left (B b + A c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 0.77 \[ 2 \, B c \sqrt {x} - \frac {2 \, {\left (3 \, B b x + 3 \, A c x + A b\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.77 \[ -\frac {2 \left (-3 B c \,x^{2}+3 A c x +3 B b x +A b \right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 27, normalized size = 0.77 \[ 2 \, B c \sqrt {x} - \frac {2 \, {\left (A b + 3 \, {\left (B b + A c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 27, normalized size = 0.77 \[ -\frac {2\,A\,b+6\,A\,c\,x+6\,B\,b\,x-6\,B\,c\,x^2}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 41, normalized size = 1.17 \[ - \frac {2 A b}{3 x^{\frac {3}{2}}} - \frac {2 A c}{\sqrt {x}} - \frac {2 B b}{\sqrt {x}} + 2 B c \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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