Optimal. Leaf size=171 \[ \frac {35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{11/2}}-\frac {35 b^2 (3 A b-2 a B)}{8 a^5 \sqrt {a+b x}}-\frac {35 b^2 (3 A b-2 a B)}{24 a^4 (a+b x)^{3/2}}-\frac {7 b (3 A b-2 a B)}{8 a^3 x (a+b x)^{3/2}}+\frac {3 A b-2 a B}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {A}{3 a x^3 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 51, 63, 208} \[ \frac {35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{11/2}}+\frac {35 \sqrt {a+b x} (3 A b-2 a B)}{12 a^4 x^2}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {35 b \sqrt {a+b x} (3 A b-2 a B)}{8 a^5 x}-\frac {A}{3 a x^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 (a+b x)^{5/2}} \, dx &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}+\frac {\left (-\frac {9 A b}{2}+3 a B\right ) \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx}{3 a}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {(7 (3 A b-2 a B)) \int \frac {1}{x^3 (a+b x)^{3/2}} \, dx}{6 a^2}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}-\frac {(35 (3 A b-2 a B)) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{6 a^3}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}+\frac {35 (3 A b-2 a B) \sqrt {a+b x}}{12 a^4 x^2}+\frac {(35 b (3 A b-2 a B)) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a^4}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}+\frac {35 (3 A b-2 a B) \sqrt {a+b x}}{12 a^4 x^2}-\frac {35 b (3 A b-2 a B) \sqrt {a+b x}}{8 a^5 x}-\frac {\left (35 b^2 (3 A b-2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^5}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}+\frac {35 (3 A b-2 a B) \sqrt {a+b x}}{12 a^4 x^2}-\frac {35 b (3 A b-2 a B) \sqrt {a+b x}}{8 a^5 x}-\frac {(35 b (3 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^5}\\ &=-\frac {A}{3 a x^3 (a+b x)^{3/2}}-\frac {3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac {7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt {a+b x}}+\frac {35 (3 A b-2 a B) \sqrt {a+b x}}{12 a^4 x^2}-\frac {35 b (3 A b-2 a B) \sqrt {a+b x}}{8 a^5 x}+\frac {35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 58, normalized size = 0.34 \[ \frac {b^2 x^3 (2 a B-3 A b) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\frac {b x}{a}+1\right )-a^3 A}{3 a^4 x^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 448, normalized size = 2.62 \[ \left [-\frac {105 \, {\left ({\left (2 \, B a b^{4} - 3 \, A b^{5}\right )} x^{5} + 2 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} + {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, A a^{5} - 105 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} - 140 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{3} - 21 \, {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2} + 6 \, {\left (2 \, B a^{5} - 3 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{48 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}}, \frac {105 \, {\left ({\left (2 \, B a b^{4} - 3 \, A b^{5}\right )} x^{5} + 2 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} + {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (8 \, A a^{5} - 105 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{4} - 140 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{3} - 21 \, {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2} + 6 \, {\left (2 \, B a^{5} - 3 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{24 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 200, normalized size = 1.17 \[ \frac {35 \, {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{5}} + \frac {210 \, {\left (b x + a\right )}^{4} B a b^{2} - 560 \, {\left (b x + a\right )}^{3} B a^{2} b^{2} + 462 \, {\left (b x + a\right )}^{2} B a^{3} b^{2} - 96 \, {\left (b x + a\right )} B a^{4} b^{2} - 16 \, B a^{5} b^{2} - 315 \, {\left (b x + a\right )}^{4} A b^{3} + 840 \, {\left (b x + a\right )}^{3} A a b^{3} - 693 \, {\left (b x + a\right )}^{2} A a^{2} b^{3} + 144 \, {\left (b x + a\right )} A a^{3} b^{3} + 16 \, A a^{4} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )}^{3} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 147, normalized size = 0.86 \[ 2 \left (-\frac {A b -B a}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4}}-\frac {4 A b -3 B a}{\sqrt {b x +a}\, a^{5}}-\frac {-\frac {35 \left (3 A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {\left (\frac {41 A b}{16}-\frac {11 B a}{8}\right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {35}{6} A a b +3 B \,a^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {55}{16} A \,a^{2} b -\frac {13}{8} B \,a^{3}\right ) \sqrt {b x +a}}{b^{3} x^{3}}}{a^{5}}\right ) b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 204, normalized size = 1.19 \[ -\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (16 \, B a^{5} - 16 \, A a^{4} b - 105 \, {\left (2 \, B a - 3 \, A b\right )} {\left (b x + a\right )}^{4} + 280 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} {\left (b x + a\right )}^{3} - 231 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{2} + 48 \, {\left (2 \, B a^{4} - 3 \, A a^{3} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {9}{2}} a^{5} b - 3 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} b + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{7} b - {\left (b x + a\right )}^{\frac {3}{2}} a^{8} b} - \frac {105 \, {\left (2 \, B a - 3 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {11}{2}} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 198, normalized size = 1.16 \[ \frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (3\,A\,b-2\,B\,a\right )}{8\,a^{11/2}}-\frac {\frac {2\,\left (A\,b^3-B\,a\,b^2\right )}{3\,a}+\frac {2\,\left (3\,A\,b^3-2\,B\,a\,b^2\right )\,\left (a+b\,x\right )}{a^2}-\frac {77\,\left (3\,A\,b^3-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^2}{8\,a^3}+\frac {35\,\left (3\,A\,b^3-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^3}{3\,a^4}-\frac {35\,\left (3\,A\,b^3-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^4}{8\,a^5}}{3\,a\,{\left (a+b\,x\right )}^{7/2}-{\left (a+b\,x\right )}^{9/2}+a^3\,{\left (a+b\,x\right )}^{3/2}-3\,a^2\,{\left (a+b\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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