Optimal. Leaf size=144 \[ -\frac {32 b \sqrt {a+b x} (8 A b-5 a B)}{15 a^5 \sqrt {x}}+\frac {16 \sqrt {a+b x} (8 A b-5 a B)}{15 a^4 x^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {16 \sqrt {a+b x} (8 A b-5 a B)}{15 a^4 x^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {32 b \sqrt {a+b x} (8 A b-5 a B)}{15 a^5 \sqrt {x}}-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx &=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-4 A b+\frac {5 a B}{2}\right )\right ) \int \frac {1}{x^{5/2} (a+b x)^{5/2}} \, dx}{5 a}\\ &=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {(2 (8 A b-5 a B)) \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}-\frac {(8 (8 A b-5 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a^3}\\ &=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}+\frac {16 (8 A b-5 a B) \sqrt {a+b x}}{15 a^4 x^{3/2}}+\frac {(16 b (8 A b-5 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^4}\\ &=-\frac {2 A}{5 a x^{5/2} (a+b x)^{3/2}}-\frac {2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac {4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt {a+b x}}+\frac {16 (8 A b-5 a B) \sqrt {a+b x}}{15 a^4 x^{3/2}}-\frac {32 b (8 A b-5 a B) \sqrt {a+b x}}{15 a^5 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 94, normalized size = 0.65 \begin {gather*} -\frac {2 \left (a^4 (3 A+5 B x)-2 a^3 b x (4 A+15 B x)+24 a^2 b^2 x^2 (2 A-5 B x)+16 a b^3 x^3 (12 A-5 B x)+128 A b^4 x^4\right )}{15 a^5 x^{5/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 106, normalized size = 0.74 \begin {gather*} \frac {2 \left (-3 a^4 A-5 a^4 B x+8 a^3 A b x+30 a^3 b B x^2-48 a^2 A b^2 x^2+120 a^2 b^2 B x^3-192 a A b^3 x^3+80 a b^3 B x^4-128 A b^4 x^4\right )}{15 a^5 x^{5/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 127, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (3 \, A a^{4} - 16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} - 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.19, size = 341, normalized size = 2.37 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {{\left (40 \, B a^{8} b^{7} - 73 \, A a^{7} b^{8}\right )} {\left (b x + a\right )}}{a^{12} b^{2} {\left | b \right |}} - \frac {5 \, {\left (17 \, B a^{9} b^{7} - 32 \, A a^{8} b^{8}\right )}}{a^{12} b^{2} {\left | b \right |}}\right )} + \frac {45 \, {\left (B a^{10} b^{7} - 2 \, A a^{9} b^{8}\right )}}{a^{12} b^{2} {\left | b \right |}}\right )}}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}}} + \frac {4 \, {\left (6 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 18 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - 9 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {9}{2}} + 8 \, B a^{3} b^{\frac {11}{2}} - 24 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} - 11 \, A a^{2} b^{\frac {13}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{4} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 0.70 \begin {gather*} -\frac {2 \left (128 A \,b^{4} x^{4}-80 B a \,b^{3} x^{4}+192 A a \,b^{3} x^{3}-120 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-30 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +5 B \,a^{4} x +3 A \,a^{4}\right )}{15 \left (b x +a \right )^{\frac {3}{2}} a^{5} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 176, normalized size = 1.22 \begin {gather*} -\frac {4 \, B b x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {32 \, A b^{2} x}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} - \frac {256 \, A b^{3} x}{15 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {2 \, B}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, B b}{3 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {16 \, A b}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} - \frac {128 \, A b^{2}}{15 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {2 \, A}{5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 129, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5\,a\,b^2}+\frac {16\,x^3\,\left (8\,A\,b-5\,B\,a\right )}{5\,a^4}+\frac {4\,x^2\,\left (8\,A\,b-5\,B\,a\right )}{5\,a^3\,b}+\frac {x^4\,\left (256\,A\,b^4-160\,B\,a\,b^3\right )}{15\,a^5\,b^2}+\frac {x\,\left (10\,B\,a^4-16\,A\,a^3\,b\right )}{15\,a^5\,b^2}\right )}{x^{9/2}+\frac {2\,a\,x^{7/2}}{b}+\frac {a^2\,x^{5/2}}{b^2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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