Optimal. Leaf size=107 \[ -\frac {(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac {3 b \sqrt {a+b x} (4 a B+A b)}{4 a}-\frac {3 b (4 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {A (a+b x)^{5/2}}{2 a x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 107, 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, 50, 63, 208} \[ -\frac {(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac {3 b \sqrt {a+b x} (4 a B+A b)}{4 a}-\frac {3 b (4 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {A (a+b x)^{5/2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx &=-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {\left (\frac {A b}{2}+2 a B\right ) \int \frac {(a+b x)^{3/2}}{x^2} \, dx}{2 a}\\ &=-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {(3 b (A b+4 a B)) \int \frac {\sqrt {a+b x}}{x} \, dx}{8 a}\\ &=\frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {1}{8} (3 b (A b+4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=\frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {1}{4} (3 (A b+4 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=\frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}-\frac {3 b (A b+4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 55, normalized size = 0.51 \[ \frac {(a+b x)^{5/2} \left (b x^2 (4 a B+A b) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x}{a}+1\right )-5 a^2 A\right )}{10 a^3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 177, normalized size = 1.65 \[ \left [\frac {3 \, {\left (4 \, B a b + A b^{2}\right )} \sqrt {a} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, B a b x^{2} - 2 \, A a^{2} - {\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt {b x + a}}{8 \, a x^{2}}, \frac {3 \, {\left (4 \, B a b + A b^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (8 \, B a b x^{2} - 2 \, A a^{2} - {\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt {b x + a}}{4 \, a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.40, size = 119, normalized size = 1.11 \[ \frac {8 \, \sqrt {b x + a} B b^{2} + \frac {3 \, {\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x + a} B a^{2} b^{2} + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{3} - 3 \, \sqrt {b x + a} A a b^{3}}{b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.79 \[ 2 \left (\sqrt {b x +a}\, B -\frac {3 \left (A b +4 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {\left (-\frac {5 A b}{8}-\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {3}{8} A a b +\frac {1}{2} B \,a^{2}\right ) \sqrt {b x +a}}{b^{2} x^{2}}\right ) b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.03, size = 130, normalized size = 1.21 \[ \frac {1}{8} \, b^{2} {\left (\frac {16 \, \sqrt {b x + a} B}{b} + \frac {3 \, {\left (4 \, B a + A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{\sqrt {a} b} - \frac {2 \, {\left ({\left (4 \, B a + 5 \, A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - {\left (4 \, B a^{2} + 3 \, A a b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{2} b - 2 \, {\left (b x + a\right )} a b + a^{2} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 127, normalized size = 1.19 \[ \frac {\left (B\,a^2\,b+\frac {3\,A\,a\,b^2}{4}\right )\,\sqrt {a+b\,x}-\left (\frac {5\,A\,b^2}{4}+B\,a\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+2\,B\,b\,\sqrt {a+b\,x}-\frac {3\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (A\,b+4\,B\,a\right )\,\sqrt {a+b\,x}}{2\,\sqrt {a}\,\left (\frac {3\,A\,b^2}{2}+6\,B\,a\,b\right )}\right )\,\left (A\,b+4\,B\,a\right )}{4\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 78.60, size = 428, normalized size = 4.00 \[ - \frac {10 A a^{3} b^{2} \sqrt {a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {6 A a^{2} b^{2} \left (a + b x\right )^{\frac {3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {3 A a^{2} b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (- a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {3 A a^{2} b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - A a b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )} + A a b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )} + \frac {2 A b^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - \frac {2 A b \sqrt {a + b x}}{x} - \frac {B a^{2} b \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {B a^{2} b \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {4 B a b \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - \frac {B a \sqrt {a + b x}}{x} + 2 B b \sqrt {a + b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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