Optimal. Leaf size=51 \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 (A c+b B)}{\sqrt{x}}+2 B c \sqrt{x} \]
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Rubi [A] time = 0.0224296, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 (A c+b B)}{\sqrt{x}}+2 B c \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac{a A}{x^{7/2}}+\frac{A b+a B}{x^{5/2}}+\frac{b B+A c}{x^{3/2}}+\frac{B c}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 a A}{5 x^{5/2}}-\frac{2 (A b+a B)}{3 x^{3/2}}-\frac{2 (b B+A c)}{\sqrt{x}}+2 B c \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.043718, size = 42, normalized size = 0.82 \[ -\frac{2 (a (3 A+5 B x)+5 x (A (b+3 c x)+3 B x (b-c x)))}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 42, normalized size = 0.8 \begin{align*} -{\frac{-30\,Bc{x}^{3}+30\,Ac{x}^{2}+30\,Bb{x}^{2}+10\,Abx+10\,aBx+6\,aA}{15}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10526, size = 54, normalized size = 1.06 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (15 \,{\left (B b + A c\right )} x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.953792, size = 100, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (15 \, B c x^{3} - 15 \,{\left (B b + A c\right )} x^{2} - 3 \, A a - 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.6039, size = 65, normalized size = 1.27 \begin{align*} - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A b}{3 x^{\frac{3}{2}}} - \frac{2 A c}{\sqrt{x}} - \frac{2 B a}{3 x^{\frac{3}{2}}} - \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 = 1.14311, size = 57, normalized size = 1.12 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (15 \, B b x^{2} + 15 \, A c x^{2} + 5 \, B a x + 5 \, A b x + 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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