Optimal. Leaf size=187 \[ -\frac {12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac {30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac {40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac {30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac {12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7} \]
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Rubi [A] time = 0.06, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac {30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac {40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac {30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac {12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 \sqrt {d+e x}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac {b^6 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (d+e x)^{3/2}}{3 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2}}{5 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2}}{9 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2}}{11 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2}}{13 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 145, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-20790 b^5 (d+e x)^5 (b d-a e)+61425 b^4 (d+e x)^4 (b d-a e)^2-100100 b^3 (d+e x)^3 (b d-a e)^3+96525 b^2 (d+e x)^2 (b d-a e)^4-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right )}{45045 e^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.14, size = 438, normalized size = 2.34 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^6 e^6+54054 a^5 b e^5 (d+e x)-90090 a^5 b d e^5+225225 a^4 b^2 d^2 e^4+96525 a^4 b^2 e^4 (d+e x)^2-270270 a^4 b^2 d e^4 (d+e x)-300300 a^3 b^3 d^3 e^3+540540 a^3 b^3 d^2 e^3 (d+e x)+100100 a^3 b^3 e^3 (d+e x)^3-386100 a^3 b^3 d e^3 (d+e x)^2+225225 a^2 b^4 d^4 e^2-540540 a^2 b^4 d^3 e^2 (d+e x)+579150 a^2 b^4 d^2 e^2 (d+e x)^2+61425 a^2 b^4 e^2 (d+e x)^4-300300 a^2 b^4 d e^2 (d+e x)^3-90090 a b^5 d^5 e+270270 a b^5 d^4 e (d+e x)-386100 a b^5 d^3 e (d+e x)^2+300300 a b^5 d^2 e (d+e x)^3+20790 a b^5 e (d+e x)^5-122850 a b^5 d e (d+e x)^4+15015 b^6 d^6-54054 b^6 d^5 (d+e x)+96525 b^6 d^4 (d+e x)^2-100100 b^6 d^3 (d+e x)^3+61425 b^6 d^2 (d+e x)^4+3003 b^6 (d+e x)^6-20790 b^6 d (d+e x)^5\right )}{45045 e^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 447, normalized size = 2.39 \begin {gather*} \frac {2 \, {\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \, {\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} - {\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 886, normalized size = 4.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 377, normalized size = 2.02 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 a^{6} e^{6}-36036 a^{5} b d \,e^{5}+51480 a^{4} b^{2} d^{2} e^{4}-45760 a^{3} b^{3} d^{3} e^{3}+24960 a^{2} b^{4} d^{4} e^{2}-7680 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{45045 e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 350, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{6} - 20790 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 61425 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 100100 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 96525 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 54054 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 162, normalized size = 0.87 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.77, size = 422, normalized size = 2.26 \begin {gather*} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (6 a b^{5} e - 6 b^{6} d\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{3 e^{6}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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