Optimal. Leaf size=177 \[ \frac{b^3 \sqrt{a+b x} (3 A b-10 a B)}{128 a^2 x}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^2 \sqrt{a+b x} (3 A b-10 a B)}{64 a x^2}+\frac{b (a+b x)^{3/2} (3 A b-10 a B)}{48 a x^3}+\frac{(a+b x)^{5/2} (3 A b-10 a B)}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5} \]
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Rubi [A] time = 0.0809073, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {78, 47, 51, 63, 208} \[ \frac{b^3 \sqrt{a+b x} (3 A b-10 a B)}{128 a^2 x}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^2 \sqrt{a+b x} (3 A b-10 a B)}{64 a x^2}+\frac{b (a+b x)^{3/2} (3 A b-10 a B)}{48 a x^3}+\frac{(a+b x)^{5/2} (3 A b-10 a B)}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} (A+B x)}{x^6} \, dx &=-\frac{A (a+b x)^{7/2}}{5 a x^5}+\frac{\left (-\frac{3 A b}{2}+5 a B\right ) \int \frac{(a+b x)^{5/2}}{x^5} \, dx}{5 a}\\ &=\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}-\frac{(b (3 A b-10 a B)) \int \frac{(a+b x)^{3/2}}{x^4} \, dx}{16 a}\\ &=\frac{b (3 A b-10 a B) (a+b x)^{3/2}}{48 a x^3}+\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}-\frac{\left (b^2 (3 A b-10 a B)\right ) \int \frac{\sqrt{a+b x}}{x^3} \, dx}{32 a}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x}}{64 a x^2}+\frac{b (3 A b-10 a B) (a+b x)^{3/2}}{48 a x^3}+\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}-\frac{\left (b^3 (3 A b-10 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{128 a}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x}}{64 a x^2}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x}}{128 a^2 x}+\frac{b (3 A b-10 a B) (a+b x)^{3/2}}{48 a x^3}+\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}+\frac{\left (b^4 (3 A b-10 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{256 a^2}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x}}{64 a x^2}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x}}{128 a^2 x}+\frac{b (3 A b-10 a B) (a+b x)^{3/2}}{48 a x^3}+\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}+\frac{\left (b^3 (3 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{128 a^2}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x}}{64 a x^2}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x}}{128 a^2 x}+\frac{b (3 A b-10 a B) (a+b x)^{3/2}}{48 a x^3}+\frac{(3 A b-10 a B) (a+b x)^{5/2}}{40 a x^4}-\frac{A (a+b x)^{7/2}}{5 a x^5}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0229107, size = 58, normalized size = 0.33 \[ -\frac{(a+b x)^{7/2} \left (7 a^5 A+b^4 x^5 (10 a B-3 A b) \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{b x}{a}+1\right )\right )}{35 a^6 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 140, normalized size = 0.8 \begin{align*} 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ({\frac{ \left ( 3\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{2}}}-{\frac{ \left ( 21\,Ab+58\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{384\,a}}+ \left ( -1/10\,Ab+1/3\,Ba \right ) \left ( bx+a \right ) ^{5/2}+{\frac{7\,a \left ( 3\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384}}-{\frac{{a}^{2} \left ( 3\,Ab-10\,Ba \right ) \sqrt{bx+a}}{256}} \right ) }-{\frac{3\,Ab-10\,Ba}{256\,{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \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 = 2.3359, size = 736, normalized size = 4.16 \begin{align*} \left [-\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt{a} x^{5} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (384 \, A a^{5} + 15 \,{\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} + 10 \,{\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{3} + 8 \,{\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3840 \, a^{3} x^{5}}, -\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (384 \, A a^{5} + 15 \,{\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} + 10 \,{\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{3} + 8 \,{\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{1920 \, a^{3} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19247, size = 281, normalized size = 1.59 \begin{align*} -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{150 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} + 580 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} - 1280 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{3} b^{5} + 700 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x + a} B a^{5} b^{5} - 45 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 210 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} + 384 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 210 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 45 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{2} b^{5} x^{5}}}{1920 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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