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 = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, 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 37
Rule 45
Rule 78
Rule 949
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*}
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Mathematica [A] time = 0.12, size = 173, normalized size = 0.92 \[ \frac {2 \sqrt {a+b x} \left (-a^3 e^3 \left (409 d^2+644 d e x+280 e^2 x^2\right )+a^2 b e^2 \left (1953 d^3+3890 d^2 e x+2632 d e^2 x^2+560 e^3 x^3\right )-a b^2 d e \left (2975 d^3+6664 d^2 e x+5168 d e^2 x^2+1344 e^3 x^3\right )+b^3 d^2 \left (1575 d^3+3850 d^2 e x+3248 d e^2 x^2+928 e^3 x^3\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 6.62, size = 487, normalized size = 2.58 \[ \frac {2 \, {\left (1575 \, b^{3} d^{5} - 2975 \, a b^{2} d^{4} e + 1953 \, a^{2} b d^{3} e^{2} - 409 \, a^{3} d^{2} e^{3} + 16 \, {\left (58 \, b^{3} d^{2} e^{3} - 84 \, a b^{2} d e^{4} + 35 \, a^{2} b e^{5}\right )} x^{3} + 8 \, {\left (406 \, b^{3} d^{3} e^{2} - 646 \, a b^{2} d^{2} e^{3} + 329 \, a^{2} b d e^{4} - 35 \, a^{3} e^{5}\right )} x^{2} + 2 \, {\left (1925 \, b^{3} d^{4} e - 3332 \, a b^{2} d^{3} e^{2} + 1945 \, a^{2} b d^{2} e^{3} - 322 \, a^{3} d e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 248, normalized size = 1.31 \[ -\frac {2 \sqrt {b x +a}\, \left (-560 a^{2} b \,e^{5} x^{3}+1344 a \,b^{2} d \,e^{4} x^{3}-928 b^{3} d^{2} e^{3} x^{3}+280 a^{3} e^{5} x^{2}-2632 a^{2} b d \,e^{4} x^{2}+5168 a \,b^{2} d^{2} e^{3} x^{2}-3248 b^{3} d^{3} e^{2} x^{2}+644 a^{3} d \,e^{4} x -3890 a^{2} b \,d^{2} e^{3} x +6664 a \,b^{2} d^{3} e^{2} x -3850 b^{3} d^{4} e x +409 a^{3} d^{2} e^{3}-1953 a^{2} b \,d^{3} e^{2}+2975 a \,b^{2} d^{4} e -1575 b^{3} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.51, size = 389, normalized size = 2.06 \[ \frac {\sqrt {d+e\,x}\,\left (\frac {-818\,a^4\,d^2\,e^3+3906\,a^3\,b\,d^3\,e^2-5950\,a^2\,b^2\,d^4\,e+3150\,a\,b^3\,d^5}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {x\,\left (-1288\,a^4\,d\,e^4+6962\,a^3\,b\,d^2\,e^3-9422\,a^2\,b^2\,d^3\,e^2+1750\,a\,b^3\,d^4\,e+3150\,b^4\,d^5\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}-\frac {x^2\,\left (560\,a^4\,e^5-3976\,a^3\,b\,d\,e^4+2556\,a^2\,b^2\,d^2\,e^3+6832\,a\,b^3\,d^3\,e^2-7700\,b^4\,d^4\,e\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {32\,b^2\,x^4\,\left (35\,a^2\,e^2-84\,a\,b\,d\,e+58\,b^2\,d^2\right )}{105\,e\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b\,x^3\,\left (35\,a^3\,e^3+161\,a^2\,b\,d\,e^2-530\,a\,b^2\,d^2\,e+406\,b^3\,d^3\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^4}\right )}{x^4\,\sqrt {a+b\,x}+\frac {d^4\,\sqrt {a+b\,x}}{e^4}+\frac {6\,d^2\,x^2\,\sqrt {a+b\,x}}{e^2}+\frac {4\,d\,x^3\,\sqrt {a+b\,x}}{e}+\frac {4\,d^3\,x\,\sqrt {a+b\,x}}{e^3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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