Optimal. Leaf size=112 \[ \frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac {b \sqrt {a+b x} (A b-6 a B)}{8 a x}-\frac {A (a+b x)^{5/2}}{3 a x^3} \]
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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 = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 47, 63, 208} \begin {gather*} \frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac {b \sqrt {a+b x} (A b-6 a B)}{8 a x}-\frac {A (a+b x)^{5/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx &=-\frac {A (a+b x)^{5/2}}{3 a x^3}+\frac {\left (-\frac {A b}{2}+3 a B\right ) \int \frac {(a+b x)^{3/2}}{x^3} \, dx}{3 a}\\ &=\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {(b (A b-6 a B)) \int \frac {\sqrt {a+b x}}{x^2} \, dx}{8 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {\left (b^2 (A b-6 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {(b (A b-6 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}+\frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 105, normalized size = 0.94 \begin {gather*} \frac {-(a+b x) \left (4 a^2 (2 A+3 B x)+2 a b x (7 A+15 B x)+3 A b^2 x^2\right )-3 b^2 x^3 \sqrt {\frac {b x}{a}+1} (6 a B-A b) \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )}{24 a x^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 118, normalized size = 1.05 \begin {gather*} \frac {\left (A b^3-6 a b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {\sqrt {a+b x} \left (18 a^3 B-3 a^2 A b-48 a^2 B (a+b x)+8 a A b (a+b x)+3 A b (a+b x)^2+30 a B (a+b x)^2\right )}{24 a b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.48, size = 210, normalized size = 1.88 \begin {gather*} \left [-\frac {3 \, {\left (6 \, 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 (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{48 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, 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 (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{24 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 145, normalized size = 1.29 \begin {gather*} \frac {\frac {3 \, {\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {30 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{3} - 48 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 18 \, \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 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 = 96, normalized size = 0.86 \begin {gather*} 2 \left (\frac {\left (A b -6 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {3}{2}}}+\frac {-\frac {\left (A b +10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a}+\left (-\frac {A b}{6}+B a \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {1}{16} A a b -\frac {3}{8} B \,a^{2}\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 = 2.00, size = 158, normalized size = 1.41 \begin {gather*} -\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (3 \, {\left (10 \, B a + A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 8 \, {\left (6 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 3 \, {\left (6 \, B a^{3} - A a^{2} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{3} a b - 3 \, {\left (b x + a\right )}^{2} a^{2} b + 3 \, {\left (b x + a\right )} a^{3} b - a^{4} b} - \frac {3 \, {\left (6 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{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 = 138, normalized size = 1.23 \begin {gather*} \frac {\left (\frac {A\,b^3}{3}-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{3/2}+\left (\frac {3\,B\,a^2\,b^2}{4}-\frac {A\,a\,b^3}{8}\right )\,\sqrt {a+b\,x}+\frac {\left (A\,b^3+10\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{5/2}}{8\,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-6\,B\,a\right )}{8\,a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 121.51, size = 806, normalized size = 7.20 \begin {gather*} - \frac {66 A a^{4} 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^{3} 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^{2} 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 {20 A a^{2} 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^{2} 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^{2} b^{3} \sqrt {\frac {1}{a^{7}}} \log {\left (a^{4} \sqrt {\frac {1}{a^{7}}} + \sqrt {a + b x} \right )}}{16} + \frac {12 A 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 a b^{3} \sqrt {\frac {1}{a^{5}}} \log {\left (- a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{4} - \frac {3 A a b^{3} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{4} - \frac {A b^{3} \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {A b^{3} \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} - \frac {A b^{2} \sqrt {a + b x}}{a x} - \frac {10 B 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 B 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 B 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 B a^{2} b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - B a b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )} + B a b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )} + \frac {2 B b^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - \frac {2 B b \sqrt {a + b x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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