Optimal. Leaf size=208 \[ -\frac{b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}+\frac{b^4 \sqrt{a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac{b^3 \sqrt{a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac{b^2 \sqrt{a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac{(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}+\frac{b \sqrt{a+b x} (7 A b-12 a B)}{160 a x^4}-\frac{A (a+b x)^{5/2}}{6 a x^6} \]
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Rubi [A] time = 0.287058, 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 \[ -\frac{b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}+\frac{b^4 \sqrt{a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac{b^3 \sqrt{a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac{b^2 \sqrt{a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac{(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}+\frac{b \sqrt{a+b x} (7 A b-12 a B)}{160 a x^4}-\frac{A (a+b x)^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/x^7,x]
[Out]
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Rubi in Sympy [A] time = 27.4312, size = 194, normalized size = 0.93 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{b \sqrt{a + b x} \left (7 A b - 12 B a\right )}{160 a x^{4}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (7 A b - 12 B a\right )}{60 a x^{5}} + \frac{b^{2} \sqrt{a + b x} \left (7 A b - 12 B a\right )}{960 a^{2} x^{3}} - \frac{b^{3} \sqrt{a + b x} \left (7 A b - 12 B a\right )}{768 a^{3} x^{2}} + \frac{b^{4} \sqrt{a + b x} \left (7 A b - 12 B a\right )}{512 a^{4} x} - \frac{b^{5} \left (7 A b - 12 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{512 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**7,x)
[Out]
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Mathematica [A] time = 0.275244, size = 148, normalized size = 0.71 \[ \frac{b^5 (12 a B-7 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{9/2}}-\frac{\sqrt{a+b x} \left (256 a^5 (5 A+6 B x)+64 a^4 b x (26 A+33 B x)+48 a^3 b^2 x^2 (A+2 B x)-8 a^2 b^3 x^3 (7 A+15 B x)+10 a b^4 x^4 (7 A+18 B x)-105 A b^5 x^5\right )}{7680 a^4 x^6} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/x^7,x]
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Maple [A] time = 0.021, size = 161, normalized size = 0.8 \[ 2\,{b}^{5} \left ({\frac{1}{{x}^{6}{b}^{6}} \left ({\frac{ \left ( 7\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1024\,{a}^{4}}}-{\frac{ \left ( 119\,Ab-204\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{3072\,{a}^{3}}}+{\frac{ \left ( 231\,Ab-396\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{2560\,{a}^{2}}}-{\frac{ \left ( 281\,Ab-116\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{2560\,a}}+ \left ( -{\frac{119\,Ab}{3072}}+{\frac{17\,Ba}{256}} \right ) \left ( bx+a \right ) ^{3/2}+{\frac{a \left ( 7\,Ab-12\,Ba \right ) \sqrt{bx+a}}{1024}} \right ) }-{\frac{7\,Ab-12\,Ba}{1024\,{a}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/x^7,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x^7,x, algorithm="maxima")
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Fricas [A] time = 0.229503, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (1280 \, A a^{5} + 15 \,{\left (12 \, B a b^{4} - 7 \, A b^{5}\right )} x^{5} - 10 \,{\left (12 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} + 8 \,{\left (12 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 48 \,{\left (44 \, B a^{4} b + A a^{3} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{5} + 13 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{15360 \, a^{\frac{9}{2}} x^{6}}, -\frac{15 \,{\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (1280 \, A a^{5} + 15 \,{\left (12 \, B a b^{4} - 7 \, A b^{5}\right )} x^{5} - 10 \,{\left (12 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} + 8 \,{\left (12 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 48 \,{\left (44 \, B a^{4} b + A a^{3} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{5} + 13 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{7680 \, \sqrt{-a} a^{4} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x^7,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*(B*x+A)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.216084, size = 324, normalized size = 1.56 \[ -\frac{\frac{15 \,{\left (12 \, B a b^{6} - 7 \, A b^{7}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{180 \,{\left (b x + a\right )}^{\frac{11}{2}} B a b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{2} b^{6} + 2376 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{3} b^{6} - 696 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{4} b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{5} b^{6} + 180 \, \sqrt{b x + a} B a^{6} b^{6} - 105 \,{\left (b x + a\right )}^{\frac{11}{2}} A b^{7} + 595 \,{\left (b x + a\right )}^{\frac{9}{2}} A a b^{7} - 1386 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{2} b^{7} + 1686 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{3} b^{7} + 595 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{4} b^{7} - 105 \, \sqrt{b x + a} A a^{5} b^{7}}{a^{4} b^{6} x^{6}}}{7680 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x^7,x, algorithm="giac")
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