Optimal. Leaf size=250 \[ -\frac {d^2 e (4 m+11) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+3) \sqrt {d^2-e^2 x^2}}+\frac {e \sqrt {d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}-\frac {3 d \sqrt {d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)}+\frac {d^3 (4 m+5) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.38, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {852, 1809, 808, 365, 364} \[ -\frac {d^2 e (4 m+11) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+3) \sqrt {d^2-e^2 x^2}}+\frac {e \sqrt {d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}+\frac {d^3 (4 m+5) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2) \sqrt {d^2-e^2 x^2}}-\frac {3 d \sqrt {d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 852
Rule 1809
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\int \frac {(g x)^m (d-e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}-\frac {\int \frac {(g x)^m \left (-d^3 e^2 (3+m)+d^2 e^3 (11+4 m) x-3 d e^4 (3+m) x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2 (3+m)}\\ &=-\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\int \frac {(g x)^m \left (d^3 e^4 (3+m) (5+4 m)-d^2 e^5 (2+m) (11+4 m) x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{e^4 (2+m) (3+m)}\\ &=-\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\left (d^3 (5+4 m)\right ) \int \frac {(g x)^m}{\sqrt {d^2-e^2 x^2}} \, dx}{2+m}-\frac {\left (d^2 e (11+4 m)\right ) \int \frac {(g x)^{1+m}}{\sqrt {d^2-e^2 x^2}} \, dx}{g (3+m)}\\ &=-\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\left (d^3 (5+4 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^m}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{(2+m) \sqrt {d^2-e^2 x^2}}-\frac {\left (d^2 e (11+4 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^{1+m}}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{g (3+m) \sqrt {d^2-e^2 x^2}}\\ &=-\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {d^3 (5+4 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (2+m) \sqrt {d^2-e^2 x^2}}-\frac {d^2 e (11+4 m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (3+m) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 245, normalized size = 0.98 \[ \frac {x \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^m \left (d^3 \left (m^3+9 m^2+26 m+24\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )-e (m+1) x \left (3 d^2 \left (m^2+7 m+12\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )+e (m+2) x \left (e (m+3) x \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};\frac {e^2 x^2}{d^2}\right )-3 d (m+4) \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\frac {e^2 x^2}{d^2}\right )\right )\right )\right )}{(m+1) (m+2) (m+3) (m+4) (d-e x) (d+e x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} \left (g x \right )^{m}}{\left (e x +d \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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