Optimal. Leaf size=267 \[ -\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}-\frac{a^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^4 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0839709, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {770, 78, 43} \[ -\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}-\frac{a^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^4 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
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Rule 770
Rule 78
Rule 43
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{x^7} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}+\frac{\left (B \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{x^6} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}+\frac{\left (B \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{a^5 b^5}{x^6}+\frac{5 a^4 b^6}{x^5}+\frac{10 a^3 b^7}{x^4}+\frac{10 a^2 b^8}{x^3}+\frac{5 a b^9}{x^2}+\frac{b^{10}}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^4 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0453606, size = 127, normalized size = 0.48 \[ -\frac{\sqrt{(a+b x)^2} \left (100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)+150 a b^4 x^4 (A+2 B x)+60 A b^5 x^5-60 b^5 B x^6 \log (x)\right )}{60 x^6 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 142, normalized size = 0.5 \begin{align*} -{\frac{-60\,B{b}^{5}\ln \left ( x \right ){x}^{6}+60\,A{x}^{5}{b}^{5}+300\,B{x}^{5}a{b}^{4}+150\,A{x}^{4}a{b}^{4}+300\,B{x}^{4}{a}^{2}{b}^{3}+200\,A{x}^{3}{a}^{2}{b}^{3}+200\,B{x}^{3}{a}^{3}{b}^{2}+150\,A{x}^{2}{a}^{3}{b}^{2}+75\,B{x}^{2}{a}^{4}b+60\,A{a}^{4}bx+12\,B{a}^{5}x+10\,A{a}^{5}}{60\, \left ( bx+a \right ) ^{5}{x}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53504, size = 270, normalized size = 1.01 \begin{align*} \frac{60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16146, size = 258, normalized size = 0.97 \begin{align*} B b^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{10 \, A a^{5} \mathrm{sgn}\left (b x + a\right ) + 60 \,{\left (5 \, B a b^{4} \mathrm{sgn}\left (b x + a\right ) + A b^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{5} + 150 \,{\left (2 \, B a^{2} b^{3} \mathrm{sgn}\left (b x + a\right ) + A a b^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{4} + 200 \,{\left (B a^{3} b^{2} \mathrm{sgn}\left (b x + a\right ) + A a^{2} b^{3} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 75 \,{\left (B a^{4} b \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 12 \,{\left (B a^{5} \mathrm{sgn}\left (b x + a\right ) + 5 \, A a^{4} b \mathrm{sgn}\left (b x + a\right )\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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