Optimal. Leaf size=376 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)} \]
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Rubi [A] time = 0.143052, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 \sqrt{d+e x} \, dx}{b^4 \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)^6 \sqrt{d+e x} \, 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)^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}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^6 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{12 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac{30 b^2 (b d-a e)^4 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac{40 b^3 (b d-a e)^3 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac{12 b^5 (b d-a e) (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac{2 b^6 (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.180859, size = 163, normalized size = 0.43 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (96525 b^2 (d+e x)^2 (b d-a e)^4-100100 b^3 (d+e x)^3 (b d-a e)^3+61425 b^4 (d+e x)^4 (b d-a e)^2-20790 b^5 (d+e x)^5 (b d-a e)-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right )}{45045 e^7 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 393, normalized size = 1.1 \begin{align*}{\frac{6006\,{x}^{6}{b}^{6}{e}^{6}+41580\,{x}^{5}a{b}^{5}{e}^{6}-5544\,{x}^{5}{b}^{6}d{e}^{5}+122850\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-37800\,{x}^{4}a{b}^{5}d{e}^{5}+5040\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+200200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-109200\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+33600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+193050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-171600\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+93600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-28800\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+108108\,x{a}^{5}b{e}^{6}-154440\,x{a}^{4}{b}^{2}d{e}^{5}+137280\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-74880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+23040\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+30030\,{a}^{6}{e}^{6}-72072\,d{e}^{5}{a}^{5}b+102960\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-91520\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+49920\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-15360\,a{b}^{5}{d}^{5}e+2048\,{b}^{6}{d}^{6}}{45045\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18362, size = 1026, normalized size = 2.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.996191, size = 1027, normalized size = 2.73 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right ) \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20111, size = 593, normalized size = 1.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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