Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{5 x^{5/2}}+2 a (2 A b+a B) \sqrt {x}+\frac {2}{7} b (A b+2 a B) x^{7/2}+\frac {2}{13} b^2 B x^{13/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 61, 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^2 A}{5 x^{5/2}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{13} b^2 B x^{13/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x^{7/2}} \, dx &=\int \left (\frac {a^2 A}{x^{7/2}}+\frac {a (2 A b+a B)}{\sqrt {x}}+b (A b+2 a B) x^{5/2}+b^2 B x^{11/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{5 x^{5/2}}+2 a (2 A b+a B) \sqrt {x}+\frac {2}{7} b (A b+2 a B) x^{7/2}+\frac {2}{13} b^2 B x^{13/2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 59, normalized size = 0.97 \begin {gather*} -\frac {2 \left (91 a^2 A-910 a A b x^3-455 a^2 B x^3-65 A b^2 x^6-130 a b B x^6-35 b^2 B x^9\right )}{455 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 54, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{2} x^{\frac {7}{2}}}{7}+\frac {4 B a b \,x^{\frac {7}{2}}}{7}+4 a b A \sqrt {x}+2 a^{2} B \sqrt {x}-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}\) | \(54\) |
default | \(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{2} x^{\frac {7}{2}}}{7}+\frac {4 B a b \,x^{\frac {7}{2}}}{7}+4 a b A \sqrt {x}+2 a^{2} B \sqrt {x}-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}\) | \(54\) |
gosper | \(-\frac {2 \left (-35 b^{2} B \,x^{9}-65 A \,b^{2} x^{6}-130 B a b \,x^{6}-910 a A b \,x^{3}-455 a^{2} B \,x^{3}+91 a^{2} A \right )}{455 x^{\frac {5}{2}}}\) | \(56\) |
trager | \(-\frac {2 \left (-35 b^{2} B \,x^{9}-65 A \,b^{2} x^{6}-130 B a b \,x^{6}-910 a A b \,x^{3}-455 a^{2} B \,x^{3}+91 a^{2} A \right )}{455 x^{\frac {5}{2}}}\) | \(56\) |
risch | \(-\frac {2 \left (-35 b^{2} B \,x^{9}-65 A \,b^{2} x^{6}-130 B a b \,x^{6}-910 a A b \,x^{3}-455 a^{2} B \,x^{3}+91 a^{2} A \right )}{455 x^{\frac {5}{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {7}{2}} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {x} - \frac {2 \, A a^{2}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (35 \, B b^{2} x^{9} + 65 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 455 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 91 \, A a^{2}\right )}}{455 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.05, size = 76, normalized size = 1.25 \begin {gather*} - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} + 4 A a b \sqrt {x} + \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + 2 B a^{2} \sqrt {x} + \frac {4 B a b x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + 2 \, B a^{2} \sqrt {x} + 4 \, A a b \sqrt {x} - \frac {2 \, A a^{2}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 51, normalized size = 0.84 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^2+4\,A\,b\,a\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}\right )-\frac {2\,A\,a^2}{5\,x^{5/2}}+\frac {2\,B\,b^2\,x^{13/2}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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