Optimal. Leaf size=155 \[ -\frac {2 a^6 A}{5 x^{5/2}}-\frac {2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac {6 a^4 b (2 a B+5 A b)}{\sqrt {x}}+10 a^3 b^2 \sqrt {x} (3 a B+4 A b)+\frac {10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+\frac {2}{7} b^5 x^{7/2} (6 a B+A b)+\frac {6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac {2}{9} b^6 B x^{9/2} \]
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Rubi [A] time = 0.08, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 76} \[ \frac {10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+10 a^3 b^2 \sqrt {x} (3 a B+4 A b)-\frac {2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac {6 a^4 b (2 a B+5 A b)}{\sqrt {x}}-\frac {2 a^6 A}{5 x^{5/2}}+\frac {6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac {2}{7} b^5 x^{7/2} (6 a B+A b)+\frac {2}{9} b^6 B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^{7/2}} \, dx\\ &=\int \left (\frac {a^6 A}{x^{7/2}}+\frac {a^5 (6 A b+a B)}{x^{5/2}}+\frac {3 a^4 b (5 A b+2 a B)}{x^{3/2}}+\frac {5 a^3 b^2 (4 A b+3 a B)}{\sqrt {x}}+5 a^2 b^3 (3 A b+4 a B) \sqrt {x}+3 a b^4 (2 A b+5 a B) x^{3/2}+b^5 (A b+6 a B) x^{5/2}+b^6 B x^{7/2}\right ) \, dx\\ &=-\frac {2 a^6 A}{5 x^{5/2}}-\frac {2 a^5 (6 A b+a B)}{3 x^{3/2}}-\frac {6 a^4 b (5 A b+2 a B)}{\sqrt {x}}+10 a^3 b^2 (4 A b+3 a B) \sqrt {x}+\frac {10}{3} a^2 b^3 (3 A b+4 a B) x^{3/2}+\frac {6}{5} a b^4 (2 A b+5 a B) x^{5/2}+\frac {2}{7} b^5 (A b+6 a B) x^{7/2}+\frac {2}{9} b^6 B x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 124, normalized size = 0.80 \[ \frac {2 \left (-21 a^6 (3 A+5 B x)-630 a^5 b x (A+3 B x)+4725 a^4 b^2 x^2 (B x-A)+2100 a^3 b^3 x^3 (3 A+B x)+315 a^2 b^4 x^4 (5 A+3 B x)+54 a b^5 x^5 (7 A+5 B x)+5 b^6 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 147, normalized size = 0.95 \[ \frac {2 \, {\left (35 \, B b^{6} x^{7} - 63 \, A a^{6} + 45 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 189 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 525 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1575 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 945 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 105 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 148, normalized size = 0.95 \[ \frac {2}{9} \, B b^{6} x^{\frac {9}{2}} + \frac {12}{7} \, B a b^{5} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{6} x^{\frac {7}{2}} + 6 \, B a^{2} b^{4} x^{\frac {5}{2}} + \frac {12}{5} \, A a b^{5} x^{\frac {5}{2}} + \frac {40}{3} \, B a^{3} b^{3} x^{\frac {3}{2}} + 10 \, A a^{2} b^{4} x^{\frac {3}{2}} + 30 \, B a^{4} b^{2} \sqrt {x} + 40 \, A a^{3} b^{3} \sqrt {x} - \frac {2 \, {\left (90 \, B a^{5} b x^{2} + 225 \, A a^{4} b^{2} x^{2} + 5 \, B a^{6} x + 30 \, A a^{5} b x + 3 \, A a^{6}\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 148, normalized size = 0.95 \[ -\frac {2 \left (-35 B \,b^{6} x^{7}-45 A \,b^{6} x^{6}-270 x^{6} B a \,b^{5}-378 A a \,b^{5} x^{5}-945 x^{5} B \,a^{2} b^{4}-1575 A \,a^{2} b^{4} x^{4}-2100 x^{4} B \,a^{3} b^{3}-6300 A \,a^{3} b^{3} x^{3}-4725 B \,a^{4} b^{2} x^{3}+4725 A \,a^{4} b^{2} x^{2}+1890 x^{2} B \,a^{5} b +630 A \,a^{5} b x +105 x B \,a^{6}+63 A \,a^{6}\right )}{315 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 148, normalized size = 0.95 \[ \frac {2}{9} \, B b^{6} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {5}{2}} + \frac {10}{3} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {3}{2}} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{6} + 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 135, normalized size = 0.87 \[ x^{7/2}\,\left (\frac {2\,A\,b^6}{7}+\frac {12\,B\,a\,b^5}{7}\right )-\frac {x\,\left (\frac {2\,B\,a^6}{3}+4\,A\,b\,a^5\right )+\frac {2\,A\,a^6}{5}+x^2\,\left (12\,B\,a^5\,b+30\,A\,a^4\,b^2\right )}{x^{5/2}}+\frac {2\,B\,b^6\,x^{9/2}}{9}+10\,a^3\,b^2\,\sqrt {x}\,\left (4\,A\,b+3\,B\,a\right )+\frac {10\,a^2\,b^3\,x^{3/2}\,\left (3\,A\,b+4\,B\,a\right )}{3}+\frac {6\,a\,b^4\,x^{5/2}\,\left (2\,A\,b+5\,B\,a\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.32, size = 204, normalized size = 1.32 \[ - \frac {2 A a^{6}}{5 x^{\frac {5}{2}}} - \frac {4 A a^{5} b}{x^{\frac {3}{2}}} - \frac {30 A a^{4} b^{2}}{\sqrt {x}} + 40 A a^{3} b^{3} \sqrt {x} + 10 A a^{2} b^{4} x^{\frac {3}{2}} + \frac {12 A a b^{5} x^{\frac {5}{2}}}{5} + \frac {2 A b^{6} x^{\frac {7}{2}}}{7} - \frac {2 B a^{6}}{3 x^{\frac {3}{2}}} - \frac {12 B a^{5} b}{\sqrt {x}} + 30 B a^{4} b^{2} \sqrt {x} + \frac {40 B a^{3} b^{3} x^{\frac {3}{2}}}{3} + 6 B a^{2} b^{4} x^{\frac {5}{2}} + \frac {12 B a b^{5} x^{\frac {7}{2}}}{7} + \frac {2 B b^{6} x^{\frac {9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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