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 = 0.0162382, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [A] time = 0.0099461, size = 28, normalized size = 0.8 \[ -\frac{2 (A (b+3 c x)+3 B x (b-c x))}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,Bc{x}^{2}+6\,Acx+6\,bBx+2\,Ab}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09719, size = 36, normalized size = 1.03 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (A b + 3 \,{\left (B b + A c\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76654, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3 \, B c x^{2} - A b - 3 \,{\left (B b + A c\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69926, size = 41, normalized size = 1.17 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11849, size = 36, normalized size = 1.03 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (3 \, B b x + 3 \, A c x + A b\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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