Optimal. Leaf size=55 \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+\frac {7}{2};\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 a d} \]
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Rubi [A] time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 365, 364} \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac {b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 372
Rubi steps
\begin {align*} \int (c+d x)^4 \left (a+b (c+d x)^2\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int x^4 \left (a+b x^2\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (\left (a+b (c+d x)^2\right )^p \left (1+\frac {b (c+d x)^2}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^4 \left (1+\frac {b x^2}{a}\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (1+\frac {b (c+d x)^2}{a}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 68, normalized size = 1.24 \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac {b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{4} {\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{4} \left (a +\left (d x +c \right )^{2} b \right )^{p}\, dx \]
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 \[ \int {\left (d x + c\right )}^{4} {\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+b\,{\left (c+d\,x\right )}^2\right )}^p\,{\left (c+d\,x\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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