Optimal. Leaf size=254 \[ \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)} \]
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Rubi [A] time = 0.35, antiderivative size = 254, 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^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)}+\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (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)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^7 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^7 \, 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)^4 (d+e x)^7}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^8}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^9}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{10}}{e^4}+\frac {b^4 (d+e x)^{11}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {b^4 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 432, normalized size = 1.70 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (495 a^4 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+220 a^3 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+66 a^2 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+12 a b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )\right )}{3960 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 4.47, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 489, normalized size = 1.93 \begin {gather*} \frac {1}{12} \, b^{4} e^{7} x^{12} + a^{4} d^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} d e^{6} + 4 \, a b^{3} e^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} d^{2} e^{5} + 28 \, a b^{3} d e^{6} + 6 \, a^{2} b^{2} e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} d^{3} e^{4} + 84 \, a b^{3} d^{2} e^{5} + 42 \, a^{2} b^{2} d e^{6} + 4 \, a^{3} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} d^{4} e^{3} + 140 \, a b^{3} d^{3} e^{4} + 126 \, a^{2} b^{2} d^{2} e^{5} + 28 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{8} + {\left (3 \, b^{4} d^{5} e^{2} + 20 \, a b^{3} d^{4} e^{3} + 30 \, a^{2} b^{2} d^{3} e^{4} + 12 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} d^{6} e + 12 \, a b^{3} d^{5} e^{2} + 30 \, a^{2} b^{2} d^{4} e^{3} + 20 \, a^{3} b d^{3} e^{4} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{7} + 28 \, a b^{3} d^{6} e + 126 \, a^{2} b^{2} d^{5} e^{2} + 140 \, a^{3} b d^{4} e^{3} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{7} + 42 \, a^{2} b^{2} d^{6} e + 84 \, a^{3} b d^{5} e^{2} + 35 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{7} + 28 \, a^{3} b d^{6} e + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{7} + 7 \, a^{4} d^{6} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 761, normalized size = 3.00 \begin {gather*} \frac {1}{12} \, b^{4} x^{12} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{11} \, b^{4} d x^{11} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {21}{10} \, b^{4} d^{2} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{9} \, b^{4} d^{3} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{8} \, b^{4} d^{4} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{4} d^{5} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{6} \, b^{4} d^{6} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{7} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{11} \, a b^{3} x^{11} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {14}{5} \, a b^{3} d x^{10} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{3} \, a b^{3} d^{2} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{2} \, a b^{3} d^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{3} d^{4} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 14 \, a b^{3} d^{5} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{5} \, a b^{3} d^{6} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{7} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a^{2} b^{2} x^{10} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {14}{3} \, a^{2} b^{2} d x^{9} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {63}{4} \, a^{2} b^{2} d^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{2} d^{3} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{2} d^{4} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {126}{5} \, a^{2} b^{2} d^{5} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {21}{2} \, a^{2} b^{2} d^{6} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{7} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, a^{3} b x^{9} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{3} b d x^{8} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b d^{2} x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {70}{3} \, a^{3} b d^{3} x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{3} b d^{4} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b d^{5} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{3} \, a^{3} b d^{6} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, a^{4} x^{8} e^{7} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{7} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{4} d^{2} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} d^{3} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{4} \, a^{4} d^{4} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} d^{5} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{4} d^{6} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{7} 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 [B] time = 0.05, size = 564, normalized size = 2.22 \begin {gather*} \frac {\left (330 b^{4} e^{7} x^{11}+1440 x^{10} a \,b^{3} e^{7}+2520 x^{10} b^{4} d \,e^{6}+2376 x^{9} a^{2} b^{2} e^{7}+11088 x^{9} a \,b^{3} d \,e^{6}+8316 x^{9} b^{4} d^{2} e^{5}+1760 x^{8} a^{3} b \,e^{7}+18480 x^{8} a^{2} b^{2} d \,e^{6}+36960 x^{8} a \,b^{3} d^{2} e^{5}+15400 x^{8} b^{4} d^{3} e^{4}+495 x^{7} a^{4} e^{7}+13860 x^{7} a^{3} b d \,e^{6}+62370 x^{7} a^{2} b^{2} d^{2} e^{5}+69300 x^{7} a \,b^{3} d^{3} e^{4}+17325 x^{7} b^{4} d^{4} e^{3}+3960 a^{4} d \,e^{6} x^{6}+47520 a^{3} b \,d^{2} e^{5} x^{6}+118800 a^{2} b^{2} d^{3} e^{4} x^{6}+79200 a \,b^{3} d^{4} e^{3} x^{6}+11880 b^{4} d^{5} e^{2} x^{6}+13860 x^{5} a^{4} d^{2} e^{5}+92400 x^{5} a^{3} b \,d^{3} e^{4}+138600 x^{5} a^{2} b^{2} d^{4} e^{3}+55440 x^{5} a \,b^{3} d^{5} e^{2}+4620 x^{5} b^{4} d^{6} e +27720 x^{4} a^{4} d^{3} e^{4}+110880 x^{4} a^{3} b \,d^{4} e^{3}+99792 x^{4} a^{2} b^{2} d^{5} e^{2}+22176 x^{4} a \,b^{3} d^{6} e +792 x^{4} b^{4} d^{7}+34650 x^{3} a^{4} d^{4} e^{3}+83160 x^{3} a^{3} b \,d^{5} e^{2}+41580 x^{3} a^{2} b^{2} d^{6} e +3960 x^{3} a \,b^{3} d^{7}+27720 x^{2} a^{4} d^{5} e^{2}+36960 x^{2} a^{3} b \,d^{6} e +7920 x^{2} a^{2} b^{2} d^{7}+13860 x \,a^{4} d^{6} e +7920 x \,a^{3} b \,d^{7}+3960 a^{4} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{3960 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.75, size = 2152, normalized size = 8.47
result too large to display
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 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^7\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
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 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{7} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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