Optimal. Leaf size=530 \[ \frac {3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac {(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac {3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac {h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+4;-n,-p;m+5;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]
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Rubi [A] time = 1.18, antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {181, 159, 140, 139, 138} \[ \frac {3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac {(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac {3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac {h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+4;-n,-p;m+5;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
Rule 159
Rule 181
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx &=\frac {h \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}+\frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}\\ &=\frac {h^2 \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+2 \frac {(h (b g-a h)) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+\frac {(b g-a h)^2 \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}\\ &=\frac {h^3 \int (a+b x)^{3+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}\right )+\frac {(b g-a h)^3 \int (a+b x)^m (c+d x)^n (e+f x)^p \, dx}{b^3}\\ &=\frac {\left (h^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{3+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}\right )+\frac {\left ((b g-a h)^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}\\ &=\frac {\left (h^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{3+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}\right )+\frac {\left ((b g-a h)^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b^3}\\ &=\frac {(b g-a h)^3 (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (1+m)}+\frac {h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (2+m;-n,-p;3+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac {h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (3+m;-n,-p;4+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}+2 \left (\frac {h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (2+m;-n,-p;3+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac {h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (3+m;-n,-p;4+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}\right )+\frac {h^3 (a+b x)^{4+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (4+m;-n,-p;5+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (4+m)}\\ \end {align*}
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Mathematica [F] time = 4.53, size = 0, normalized size = 0.00 \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 3.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right )^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p}\, dx \]
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 \[ \int {\left (h x + g\right )}^{3} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e+f\,x\right )}^p\,{\left (g+h\,x\right )}^3\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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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