Optimal. Leaf size=107 \[ -\frac {2 a^4 A}{3 x^{3/2}}-\frac {2 a^3 (a B+4 A b)}{\sqrt {x}}+4 a^2 b \sqrt {x} (2 a B+3 A b)+\frac {2}{5} b^3 x^{5/2} (4 a B+A b)+\frac {4}{3} a b^2 x^{3/2} (3 a B+2 A b)+\frac {2}{7} b^4 B x^{7/2} \]
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Rubi [A] time = 0.05, antiderivative size = 107, 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} \[ -\frac {2 a^3 (a B+4 A b)}{\sqrt {x}}+4 a^2 b \sqrt {x} (2 a B+3 A b)-\frac {2 a^4 A}{3 x^{3/2}}+\frac {4}{3} a b^2 x^{3/2} (3 a B+2 A b)+\frac {2}{5} b^3 x^{5/2} (4 a B+A b)+\frac {2}{7} b^4 B x^{7/2} \]
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 )^2}{x^{5/2}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^{5/2}} \, dx\\ &=\int \left (\frac {a^4 A}{x^{5/2}}+\frac {a^3 (4 A b+a B)}{x^{3/2}}+\frac {2 a^2 b (3 A b+2 a B)}{\sqrt {x}}+2 a b^2 (2 A b+3 a B) \sqrt {x}+b^3 (A b+4 a B) x^{3/2}+b^4 B x^{5/2}\right ) \, dx\\ &=-\frac {2 a^4 A}{3 x^{3/2}}-\frac {2 a^3 (4 A b+a B)}{\sqrt {x}}+4 a^2 b (3 A b+2 a B) \sqrt {x}+\frac {4}{3} a b^2 (2 A b+3 a B) x^{3/2}+\frac {2}{5} b^3 (A b+4 a B) x^{5/2}+\frac {2}{7} b^4 B x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 86, normalized size = 0.80 \[ \frac {-70 a^4 (A+3 B x)+840 a^3 b x (B x-A)+420 a^2 b^2 x^2 (3 A+B x)+56 a b^3 x^3 (5 A+3 B x)+6 b^4 x^4 (7 A+5 B x)}{105 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 99, normalized size = 0.93 \[ \frac {2 \, {\left (15 \, B b^{4} x^{5} - 35 \, A a^{4} + 21 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 70 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 105 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{105 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 99, normalized size = 0.93 \[ \frac {2}{7} \, B b^{4} x^{\frac {7}{2}} + \frac {8}{5} \, B a b^{3} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{4} x^{\frac {5}{2}} + 4 \, B a^{2} b^{2} x^{\frac {3}{2}} + \frac {8}{3} \, A a b^{3} x^{\frac {3}{2}} + 8 \, B a^{3} b \sqrt {x} + 12 \, A a^{2} b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{4} x + 12 \, A a^{3} b x + A a^{4}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 0.93 \[ -\frac {2 \left (-15 b^{4} B \,x^{5}-21 A \,b^{4} x^{4}-84 x^{4} B a \,b^{3}-140 A a \,b^{3} x^{3}-210 B \,a^{2} b^{2} x^{3}-630 A \,a^{2} b^{2} x^{2}-420 B \,a^{3} b \,x^{2}+420 A \,a^{3} b x +105 B \,a^{4} x +35 A \,a^{4}\right )}{105 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 99, normalized size = 0.93 \[ \frac {2}{7} \, B b^{4} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {5}{2}} + \frac {4}{3} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {3}{2}} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 92, normalized size = 0.86 \[ x^{5/2}\,\left (\frac {2\,A\,b^4}{5}+\frac {8\,B\,a\,b^3}{5}\right )-\frac {x\,\left (2\,B\,a^4+8\,A\,b\,a^3\right )+\frac {2\,A\,a^4}{3}}{x^{3/2}}+\frac {2\,B\,b^4\,x^{7/2}}{7}+4\,a^2\,b\,\sqrt {x}\,\left (3\,A\,b+2\,B\,a\right )+\frac {4\,a\,b^2\,x^{3/2}\,\left (2\,A\,b+3\,B\,a\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.16, size = 139, normalized size = 1.30 \[ - \frac {2 A a^{4}}{3 x^{\frac {3}{2}}} - \frac {8 A a^{3} b}{\sqrt {x}} + 12 A a^{2} b^{2} \sqrt {x} + \frac {8 A a b^{3} x^{\frac {3}{2}}}{3} + \frac {2 A b^{4} x^{\frac {5}{2}}}{5} - \frac {2 B a^{4}}{\sqrt {x}} + 8 B a^{3} b \sqrt {x} + 4 B a^{2} b^{2} x^{\frac {3}{2}} + \frac {8 B a b^{3} x^{\frac {5}{2}}}{5} + \frac {2 B b^{4} x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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