Optimal. Leaf size=264 \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)}+\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)} \]
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Rubi [A] time = 0.127918, antiderivative size = 264, 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} \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)}+\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^{5/2} \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)^{5/2} \, 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)^{5/2} \, 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)^{5/2}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac{b^4 (d+e x)^{13/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^4 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac{8 b (b d-a e)^3 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac{12 b^2 (b d-a e)^2 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac{8 b^3 (b d-a e) (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac{2 b^4 (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.125635, size = 172, normalized size = 0.65 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+2860 a^3 b e^3 (7 e x-2 d)+6435 a^4 e^4+60 a b^3 e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 202, normalized size = 0.8 \begin{align*}{\frac{6006\,{x}^{4}{b}^{4}{e}^{4}+27720\,{x}^{3}a{b}^{3}{e}^{4}-3696\,{x}^{3}{b}^{4}d{e}^{3}+49140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-15120\,{x}^{2}a{b}^{3}d{e}^{3}+2016\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40040\,x{a}^{3}b{e}^{4}-21840\,x{a}^{2}{b}^{2}d{e}^{3}+6720\,xa{b}^{3}{d}^{2}{e}^{2}-896\,x{b}^{4}{d}^{3}e+12870\,{a}^{4}{e}^{4}-11440\,d{e}^{3}{a}^{3}b+6240\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1920\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{45045\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19691, size = 799, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \,{\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} +{\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} a}{3003 \, e^{4}} + \frac{2 \,{\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \,{\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \,{\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d} b}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04813, size = 855, normalized size = 3.24 \begin{align*} \frac{2 \,{\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \,{\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \,{\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} -{\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27783, size = 1278, normalized size = 4.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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