Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{7} a b (A b+a B) x^{7/2}+\frac {2}{11} b^2 (A b+3 a B) x^{11/2}+\frac {2}{15} b^3 B x^{15/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459}
\begin {gather*} -\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{3} a^2 x^{3/2} (a B+3 A b)+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{15} b^3 B x^{15/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac {a^3 A}{x^{3/2}}+a^2 (3 A b+a B) \sqrt {x}+3 a b (A b+a B) x^{5/2}+b^2 (A b+3 a B) x^{9/2}+b^3 B x^{13/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{7} a b (A b+a B) x^{7/2}+\frac {2}{11} b^2 (A b+3 a B) x^{11/2}+\frac {2}{15} b^3 B x^{15/2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 83, normalized size = 1.00 \begin {gather*} -\frac {2 \left (1155 a^3 A-1155 a^2 A b x^2-385 a^3 B x^2-495 a A b^2 x^4-495 a^2 b B x^4-105 A b^3 x^6-315 a b^2 B x^6-77 b^3 B x^8\right )}{1155 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 78, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {15}{2}}}{15}+\frac {2 A \,b^{3} x^{\frac {11}{2}}}{11}+\frac {6 B a \,b^{2} x^{\frac {11}{2}}}{11}+\frac {6 A a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {6 B \,a^{2} b \,x^{\frac {7}{2}}}{7}+2 A \,a^{2} b \,x^{\frac {3}{2}}+\frac {2 B \,a^{3} x^{\frac {3}{2}}}{3}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(78\) |
default | \(\frac {2 b^{3} B \,x^{\frac {15}{2}}}{15}+\frac {2 A \,b^{3} x^{\frac {11}{2}}}{11}+\frac {6 B a \,b^{2} x^{\frac {11}{2}}}{11}+\frac {6 A a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {6 B \,a^{2} b \,x^{\frac {7}{2}}}{7}+2 A \,a^{2} b \,x^{\frac {3}{2}}+\frac {2 B \,a^{3} x^{\frac {3}{2}}}{3}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(78\) |
gosper | \(-\frac {2 \left (-77 B \,b^{3} x^{8}-105 x^{6} A \,b^{3}-315 x^{6} B a \,b^{2}-495 A a \,b^{2} x^{4}-495 x^{4} B \,a^{2} b -1155 x^{2} A \,a^{2} b -385 B \,a^{3} x^{2}+1155 A \,a^{3}\right )}{1155 \sqrt {x}}\) | \(80\) |
trager | \(-\frac {2 \left (-77 B \,b^{3} x^{8}-105 x^{6} A \,b^{3}-315 x^{6} B a \,b^{2}-495 A a \,b^{2} x^{4}-495 x^{4} B \,a^{2} b -1155 x^{2} A \,a^{2} b -385 B \,a^{3} x^{2}+1155 A \,a^{3}\right )}{1155 \sqrt {x}}\) | \(80\) |
risch | \(-\frac {2 \left (-77 B \,b^{3} x^{8}-105 x^{6} A \,b^{3}-315 x^{6} B a \,b^{2}-495 A a \,b^{2} x^{4}-495 x^{4} B \,a^{2} b -1155 x^{2} A \,a^{2} b -385 B \,a^{3} x^{2}+1155 A \,a^{3}\right )}{1155 \sqrt {x}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {11}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (77 \, B b^{3} x^{8} + 105 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 495 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} - 1155 \, A a^{3} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{1155 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 110, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{3}}{\sqrt {x}} + 2 A a^{2} b x^{\frac {3}{2}} + \frac {6 A a b^{2} x^{\frac {7}{2}}}{7} + \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {6 B a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 B a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {6}{11} \, B a b^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 69, normalized size = 0.83 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^3}{3}+2\,A\,b\,a^2\right )+x^{11/2}\,\left (\frac {2\,A\,b^3}{11}+\frac {6\,B\,a\,b^2}{11}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{15/2}}{15}+\frac {6\,a\,b\,x^{7/2}\,\left (A\,b+B\,a\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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