Optimal. Leaf size=208 \[ -\frac{b^3 \sqrt{a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac{b^2 \sqrt{a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac{b^4 \sqrt{a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac{b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}+\frac{b \sqrt{a+b x} (7 A b-12 a B)}{160 a x^4}+\frac{(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6} \]
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Rubi [A] time = 0.0999218, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{b^3 \sqrt{a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac{b^2 \sqrt{a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac{b^4 \sqrt{a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac{b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}+\frac{b \sqrt{a+b x} (7 A b-12 a B)}{160 a x^4}+\frac{(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6} \]
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)^{3/2} (A+B x)}{x^7} \, dx &=-\frac{A (a+b x)^{5/2}}{6 a x^6}+\frac{\left (-\frac{7 A b}{2}+6 a B\right ) \int \frac{(a+b x)^{3/2}}{x^6} \, dx}{6 a}\\ &=\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}-\frac{(b (7 A b-12 a B)) \int \frac{\sqrt{a+b x}}{x^5} \, dx}{40 a}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}-\frac{\left (b^2 (7 A b-12 a B)\right ) \int \frac{1}{x^4 \sqrt{a+b x}} \, dx}{320 a}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{b^2 (7 A b-12 a B) \sqrt{a+b x}}{960 a^2 x^3}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}+\frac{\left (b^3 (7 A b-12 a B)\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{384 a^2}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{b^2 (7 A b-12 a B) \sqrt{a+b x}}{960 a^2 x^3}-\frac{b^3 (7 A b-12 a B) \sqrt{a+b x}}{768 a^3 x^2}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}-\frac{\left (b^4 (7 A b-12 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{512 a^3}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{b^2 (7 A b-12 a B) \sqrt{a+b x}}{960 a^2 x^3}-\frac{b^3 (7 A b-12 a B) \sqrt{a+b x}}{768 a^3 x^2}+\frac{b^4 (7 A b-12 a B) \sqrt{a+b x}}{512 a^4 x}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}+\frac{\left (b^5 (7 A b-12 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{1024 a^4}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{b^2 (7 A b-12 a B) \sqrt{a+b x}}{960 a^2 x^3}-\frac{b^3 (7 A b-12 a B) \sqrt{a+b x}}{768 a^3 x^2}+\frac{b^4 (7 A b-12 a B) \sqrt{a+b x}}{512 a^4 x}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}+\frac{\left (b^4 (7 A b-12 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{512 a^4}\\ &=\frac{b (7 A b-12 a B) \sqrt{a+b x}}{160 a x^4}+\frac{b^2 (7 A b-12 a B) \sqrt{a+b x}}{960 a^2 x^3}-\frac{b^3 (7 A b-12 a B) \sqrt{a+b x}}{768 a^3 x^2}+\frac{b^4 (7 A b-12 a B) \sqrt{a+b x}}{512 a^4 x}+\frac{(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac{A (a+b x)^{5/2}}{6 a x^6}-\frac{b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0228278, size = 58, normalized size = 0.28 \[ -\frac{(a+b x)^{5/2} \left (5 a^6 A+b^5 x^6 (7 A b-12 a B) \, _2F_1\left (\frac{5}{2},6;\frac{7}{2};\frac{b x}{a}+1\right )\right )}{30 a^7 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 161, normalized size = 0.8 \begin{align*} 2\,{b}^{5} \left ({\frac{1}{{b}^{6}{x}^{6}} \left ({\frac{ \left ( 7\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1024\,{a}^{4}}}-{\frac{ \left ( 119\,Ab-204\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{3072\,{a}^{3}}}+{\frac{ \left ( 231\,Ab-396\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{2560\,{a}^{2}}}-{\frac{ \left ( 281\,Ab-116\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{2560\,a}}+ \left ( -{\frac{119\,Ab}{3072}}+{\frac{17\,Ba}{256}} \right ) \left ( bx+a \right ) ^{3/2}+{\frac{a \left ( 7\,Ab-12\,Ba \right ) \sqrt{bx+a}}{1024}} \right ) }-{\frac{7\,Ab-12\,Ba}{1024\,{a}^{9/2}}{\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 = 2.3769, size = 832, normalized size = 4. \begin{align*} \left [-\frac{15 \,{\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} \sqrt{a} x^{6} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (1280 \, A a^{6} + 15 \,{\left (12 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} - 10 \,{\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \,{\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \,{\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{15360 \, a^{5} x^{6}}, -\frac{15 \,{\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (1280 \, A a^{6} + 15 \,{\left (12 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} - 10 \,{\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \,{\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \,{\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{7680 \, a^{5} x^{6}}\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 = 1.24057, size = 324, normalized size = 1.56 \begin{align*} -\frac{\frac{15 \,{\left (12 \, B a b^{6} - 7 \, A b^{7}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{180 \,{\left (b x + a\right )}^{\frac{11}{2}} B a b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{2} b^{6} + 2376 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{3} b^{6} - 696 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{4} b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{5} b^{6} + 180 \, \sqrt{b x + a} B a^{6} b^{6} - 105 \,{\left (b x + a\right )}^{\frac{11}{2}} A b^{7} + 595 \,{\left (b x + a\right )}^{\frac{9}{2}} A a b^{7} - 1386 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{2} b^{7} + 1686 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{3} b^{7} + 595 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{4} b^{7} - 105 \, \sqrt{b x + a} A a^{5} b^{7}}{a^{4} b^{6} x^{6}}}{7680 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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