Optimal. Leaf size=112 \[ -\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {b \sqrt {a+b x} (A b-2 a B)}{8 a^2 x}+\frac {\sqrt {a+b x} (A b-2 a B)}{4 a x^2}-\frac {A (a+b x)^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \begin {gather*} -\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {b \sqrt {a+b x} (A b-2 a B)}{8 a^2 x}+\frac {\sqrt {a+b x} (A b-2 a B)}{4 a x^2}-\frac {A (a+b x)^{3/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^4} \, dx &=-\frac {A (a+b x)^{3/2}}{3 a x^3}+\frac {\left (-\frac {3 A b}{2}+3 a B\right ) \int \frac {\sqrt {a+b x}}{x^3} \, dx}{3 a}\\ &=\frac {(A b-2 a B) \sqrt {a+b x}}{4 a x^2}-\frac {A (a+b x)^{3/2}}{3 a x^3}-\frac {(b (A b-2 a B)) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a}\\ &=\frac {(A b-2 a B) \sqrt {a+b x}}{4 a x^2}+\frac {b (A b-2 a B) \sqrt {a+b x}}{8 a^2 x}-\frac {A (a+b x)^{3/2}}{3 a x^3}+\frac {\left (b^2 (A b-2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^2}\\ &=\frac {(A b-2 a B) \sqrt {a+b x}}{4 a x^2}+\frac {b (A b-2 a B) \sqrt {a+b x}}{8 a^2 x}-\frac {A (a+b x)^{3/2}}{3 a x^3}+\frac {(b (A b-2 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a^2}\\ &=\frac {(A b-2 a B) \sqrt {a+b x}}{4 a x^2}+\frac {b (A b-2 a B) \sqrt {a+b x}}{8 a^2 x}-\frac {A (a+b x)^{3/2}}{3 a x^3}-\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 57, normalized size = 0.51 \begin {gather*} -\frac {(a+b x)^{3/2} \left (a^3 A+b^2 x^3 (2 a B-A b) \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x}{a}+1\right )\right )}{3 a^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 108, normalized size = 0.96 \begin {gather*} \frac {\left (2 a b^2 B-A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {\sqrt {a+b x} \left (6 a^3 B-3 a^2 A b-8 a A b (a+b x)+3 A b (a+b x)^2-6 a B (a+b x)^2\right )}{24 a^2 b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 209, normalized size = 1.87 \begin {gather*} \left [-\frac {3 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, A a^{3} + 3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt {b x + a}}{48 \, a^{3} x^{3}}, -\frac {3 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (8 \, A a^{3} + 3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt {b x + a}}{24 \, a^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 128, normalized size = 1.14 \begin {gather*} -\frac {\frac {3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {6 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{3} - 6 \, \sqrt {b x + a} B a^{3} b^{3} - 3 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{4} + 8 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{4} + 3 \, \sqrt {b x + a} A a^{2} b^{4}}{a^{2} b^{3} x^{3}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 91, normalized size = 0.81 \begin {gather*} 2 \left (-\frac {\left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}+\frac {-\frac {\left (b x +a \right )^{\frac {3}{2}} A b}{6 a}+\frac {\left (A b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}+\left (-\frac {A b}{16}+\frac {B a}{8}\right ) \sqrt {b x +a}}{b^{3} x^{3}}\right ) b^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 152, normalized size = 1.36 \begin {gather*} -\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (8 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b + 3 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 3 \, {\left (2 \, B a^{3} - A a^{2} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{3} a^{2} b - 3 \, {\left (b x + a\right )}^{2} a^{3} b + 3 \, {\left (b x + a\right )} a^{4} b - a^{5} b} + \frac {3 \, {\left (2 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 129, normalized size = 1.15 \begin {gather*} \frac {\left (\frac {A\,b^3}{8}-\frac {B\,a\,b^2}{4}\right )\,\sqrt {a+b\,x}-\frac {\left (A\,b^3-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{5/2}}{8\,a^2}+\frac {A\,b^3\,{\left (a+b\,x\right )}^{3/2}}{3\,a}}{3\,a\,{\left (a+b\,x\right )}^2-3\,a^2\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^3+a^3}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-2\,B\,a\right )}{8\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 38.71, size = 666, normalized size = 5.95 \begin {gather*} - \frac {66 A a^{3} b^{3} \sqrt {a + b x}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} + \frac {80 A a^{2} b^{3} \left (a + b x\right )^{\frac {3}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} - \frac {30 A a b^{3} \left (a + b x\right )^{\frac {5}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} - \frac {10 A a b^{3} \sqrt {a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} - \frac {5 A a b^{3} \sqrt {\frac {1}{a^{7}}} \log {\left (- a^{4} \sqrt {\frac {1}{a^{7}}} + \sqrt {a + b x} \right )}}{16} + \frac {5 A a b^{3} \sqrt {\frac {1}{a^{7}}} \log {\left (a^{4} \sqrt {\frac {1}{a^{7}}} + \sqrt {a + b x} \right )}}{16} + \frac {6 A b^{3} \left (a + b x\right )^{\frac {3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {3 A b^{3} \sqrt {\frac {1}{a^{5}}} \log {\left (- a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {3 A b^{3} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {10 B a^{2} b^{2} \sqrt {a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {6 B a b^{2} \left (a + b x\right )^{\frac {3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {3 B a b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (- a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {3 B a b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {B b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {B b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} - \frac {B b \sqrt {a + b x}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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