Optimal. Leaf size=189 \[ \frac{32 b \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 \sqrt{d+e x} (b d-a e)^4}+\frac{16 \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 (d+e x)^{3/2} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{7 (d+e x)^{7/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (23 b d-14 a e)}{35 (d+e x)^{5/2} (b d-a e)^2} \]
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Rubi [A] time = 0.19222, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {949, 78, 45, 37} \[ \frac{32 b \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 \sqrt{d+e x} (b d-a e)^4}+\frac{16 \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 (d+e x)^{3/2} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{7 (d+e x)^{7/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (23 b d-14 a e)}{35 (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 949
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{2 \int \frac{3 d (17 b d-14 a e)+28 e (b d-a e) x}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{\left (8 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^2}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{16 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^3 (d+e x)^{3/2}}+\frac{\left (16 b \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{105 (b d-a e)^3}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{16 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^3 (d+e x)^{3/2}}+\frac{32 b \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^4 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.124592, size = 173, normalized size = 0.92 \[ \frac{2 \sqrt{a+b x} \left (a^2 b e^2 \left (3890 d^2 e x+1953 d^3+2632 d e^2 x^2+560 e^3 x^3\right )-a^3 e^3 \left (409 d^2+644 d e x+280 e^2 x^2\right )-a b^2 d e \left (6664 d^2 e x+2975 d^3+5168 d e^2 x^2+1344 e^3 x^3\right )+b^3 d^2 \left (3850 d^2 e x+1575 d^3+3248 d e^2 x^2+928 e^3 x^3\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 248, normalized size = 1.3 \begin{align*} -{\frac{-1120\,{a}^{2}b{e}^{5}{x}^{3}+2688\,a{b}^{2}d{e}^{4}{x}^{3}-1856\,{b}^{3}{d}^{2}{e}^{3}{x}^{3}+560\,{a}^{3}{e}^{5}{x}^{2}-5264\,{a}^{2}bd{e}^{4}{x}^{2}+10336\,a{b}^{2}{d}^{2}{e}^{3}{x}^{2}-6496\,{b}^{3}{d}^{3}{e}^{2}{x}^{2}+1288\,{a}^{3}d{e}^{4}x-7780\,{a}^{2}b{d}^{2}{e}^{3}x+13328\,a{b}^{2}{d}^{3}{e}^{2}x-7700\,{b}^{3}{d}^{4}ex+818\,{a}^{3}{d}^{2}{e}^{3}-3906\,{a}^{2}b{d}^{3}{e}^{2}+5950\,a{b}^{2}{d}^{4}e-3150\,{b}^{3}{d}^{5}}{105\,{e}^{4}{a}^{4}-420\,b{e}^{3}d{a}^{3}+630\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-420\,a{b}^{3}{d}^{3}e+105\,{b}^{4}{d}^{4}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 14.2424, size = 1021, normalized size = 5.4 \begin{align*} \frac{2 \,{\left (1575 \, b^{3} d^{5} - 2975 \, a b^{2} d^{4} e + 1953 \, a^{2} b d^{3} e^{2} - 409 \, a^{3} d^{2} e^{3} + 16 \,{\left (58 \, b^{3} d^{2} e^{3} - 84 \, a b^{2} d e^{4} + 35 \, a^{2} b e^{5}\right )} x^{3} + 8 \,{\left (406 \, b^{3} d^{3} e^{2} - 646 \, a b^{2} d^{2} e^{3} + 329 \, a^{2} b d e^{4} - 35 \, a^{3} e^{5}\right )} x^{2} + 2 \,{\left (1925 \, b^{3} d^{4} e - 3332 \, a b^{2} d^{3} e^{2} + 1945 \, a^{2} b d^{2} e^{3} - 322 \, a^{3} d e^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \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 = 1.45319, size = 707, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (58 \, b^{10} d^{2} e^{6} - 84 \, a b^{9} d e^{7} + 35 \, a^{2} b^{8} e^{8}\right )}{\left (b x + a\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}} + \frac{7 \,{\left (58 \, b^{11} d^{3} e^{5} - 142 \, a b^{10} d^{2} e^{6} + 119 \, a^{2} b^{9} d e^{7} - 35 \, a^{3} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )} + \frac{35 \,{\left (55 \, b^{12} d^{4} e^{4} - 188 \, a b^{11} d^{3} e^{5} + 243 \, a^{2} b^{10} d^{2} e^{6} - 142 \, a^{3} b^{9} d e^{7} + 32 \, a^{4} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )}{\left (b x + a\right )} + \frac{105 \,{\left (15 \, b^{13} d^{5} e^{3} - 65 \, a b^{12} d^{4} e^{4} + 113 \, a^{2} b^{11} d^{3} e^{5} - 99 \, a^{3} b^{10} d^{2} e^{6} + 44 \, a^{4} b^{9} d e^{7} - 8 \, a^{5} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )} \sqrt{b x + a}}{105 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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