Optimal. Leaf size=150 \[ \frac {3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}+\frac {d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}-\frac {2 e x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {5}{2},2-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3} \]
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Rubi [A] time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {852, 1652, 446, 77, 12, 365, 364} \[ -\frac {2 e x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {5}{2},2-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3}+\frac {d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}+\frac {3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 77
Rule 364
Rule 365
Rule 446
Rule 852
Rule 1652
Rubi steps
\begin {align*} \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\int x^3 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\int -2 d e x^4 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^3 \left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x \left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right ) \, dx,x,x^2\right )-(2 d e) \int x^4 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2 d^4 \left (d^2-e^2 x\right )^{-2+p}}{e^2}-\frac {3 d^2 \left (d^2-e^2 x\right )^{-1+p}}{e^2}+\frac {\left (d^2-e^2 x\right )^p}{e^2}\right ) \, dx,x,x^2\right )-\frac {\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3}\\ &=\frac {d^4 \left (d^2-e^2 x^2\right )^{-1+p}}{e^4 (1-p)}+\frac {3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (1+p)}-\frac {2 e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},2-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3}\\ \end {align*}
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Mathematica [B] time = 0.29, size = 332, normalized size = 2.21 \[ \frac {2^{p-2} \left (\frac {e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (-8 d e (p+1) x \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-6 d (d-e x) \left (1-\frac {e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac {d-e x}{2 d}\right )+d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac {d-e x}{2 d}\right )-d e x \left (1-\frac {e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac {d-e x}{2 d}\right )-2 d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p+2 e^2 x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p+2 d^2 \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p\right )}{e^4 (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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