Optimal. Leaf size=146 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]
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Rubi [A] time = 0.13, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^4 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^4 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2 (d+e x)^4}{e^2}-\frac {2 b (b d-a e) (d+e x)^5}{e^2}+\frac {b^2 (d+e x)^6}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^2 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac {b (b d-a e) (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}+\frac {b^2 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 163, normalized size = 1.12 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (21 a^2 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+7 a b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )}{105 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.62, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 156, normalized size = 1.07 \begin {gather*} \frac {1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} + {\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 254, normalized size = 1.74 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{5} \, a b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + a b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{2} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{4} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 189, normalized size = 1.29 \begin {gather*} \frac {\left (15 b^{2} e^{4} x^{6}+35 x^{5} a b \,e^{4}+70 x^{5} b^{2} d \,e^{3}+21 x^{4} a^{2} e^{4}+168 x^{4} a b d \,e^{3}+126 x^{4} b^{2} d^{2} e^{2}+105 a^{2} d \,e^{3} x^{3}+315 a b \,d^{2} e^{2} x^{3}+105 b^{2} d^{3} e \,x^{3}+210 x^{2} a^{2} d^{2} e^{2}+280 x^{2} a b \,d^{3} e +35 x^{2} b^{2} d^{4}+210 a^{2} d^{3} e x +105 a b \,d^{4} x +105 a^{2} d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{105 b x +105 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 996, normalized size = 6.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 1095, normalized size = 7.50 \begin {gather*} a\,d^4\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^4\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^4\,x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{7\,b}-\frac {a^3\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{24\,b^5}+\frac {a\,e^4\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}+\frac {2\,d\,e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{3\,b}-\frac {11\,a\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^5+5\,b^3\,x^3\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-14\,a^3\,b^2\,x^2-13\,a^4\,b\,x-9\,a\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )+12\,a^2\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{210\,b^5}+\frac {6\,d^2\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}+\frac {d^3\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{b}-\frac {29\,a^2\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{280\,b^5}-\frac {a\,d^3\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^4}-\frac {3\,a^3\,d^2\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b^2}-\frac {7\,a\,d^2\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{10\,b^3}-\frac {19\,a^2\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{30\,b^4}-\frac {a^3\,d\,e^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{15\,b^6}+\frac {3\,a\,d^2\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b^2}+\frac {4\,a\,d\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}-\frac {a^2\,d^3\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b}-\frac {3\,a\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{10\,b^4}-\frac {33\,a^2\,d^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{80\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 168, normalized size = 1.15 \begin {gather*} a^{2} d^{4} x + \frac {b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a b e^{4}}{3} + \frac {2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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