Optimal. Leaf size=189 \[ \frac {6 d^2 \sqrt {a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac {4 d (23 b d-14 a e) \sqrt {a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac {16 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt {a+b x}}{105 (b d-a e)^3 (d+e x)^{3/2}}+\frac {32 b \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt {a+b x}}{105 (b d-a e)^4 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.12, 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 = {963, 79, 47, 37}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 963
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 = 0.16, size = 185, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {a+b x} \left (1575 b^3 d^2-2100 a b^2 d e+840 a^2 b e^2-\frac {45 d^2 e^3 (a+b x)^3}{(d+e x)^3}+\frac {273 b d^2 e^2 (a+b x)^2}{(d+e x)^2}-\frac {84 a d e^3 (a+b x)^2}{(d+e x)^2}-\frac {875 b^2 d^2 e (a+b x)}{d+e x}+\frac {840 a b d e^2 (a+b x)}{d+e x}-\frac {280 a^2 e^3 (a+b x)}{d+e x}\right )}{105 (b d-a e)^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 207, normalized size = 1.10
method | result | size |
default | \(-\frac {2 \sqrt {b x +a}\, \left (-560 a^{2} b \,e^{5} x^{3}+1344 a \,b^{2} d \,e^{4} x^{3}-928 b^{3} d^{2} e^{3} x^{3}+280 a^{3} e^{5} x^{2}-2632 a^{2} b d \,e^{4} x^{2}+5168 a \,b^{2} d^{2} e^{3} x^{2}-3248 b^{3} d^{3} e^{2} x^{2}+644 a^{3} d \,e^{4} x -3890 a^{2} b \,d^{2} e^{3} x +6664 a \,b^{2} d^{3} e^{2} x -3850 b^{3} d^{4} e x +409 a^{3} d^{2} e^{3}-1953 a^{2} b \,d^{3} e^{2}+2975 a \,b^{2} d^{4} e -1575 b^{3} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (a e -b d \right )^{4}}\) | \(207\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-560 a^{2} b \,e^{5} x^{3}+1344 a \,b^{2} d \,e^{4} x^{3}-928 b^{3} d^{2} e^{3} x^{3}+280 a^{3} e^{5} x^{2}-2632 a^{2} b d \,e^{4} x^{2}+5168 a \,b^{2} d^{2} e^{3} x^{2}-3248 b^{3} d^{3} e^{2} x^{2}+644 a^{3} d \,e^{4} x -3890 a^{2} b \,d^{2} e^{3} x +6664 a \,b^{2} d^{3} e^{2} x -3850 b^{3} d^{4} e x +409 a^{3} d^{2} e^{3}-1953 a^{2} b \,d^{3} e^{2}+2975 a \,b^{2} d^{4} e -1575 b^{3} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(248\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 482 vs.
\(2 (174) = 348\).
time = 8.21, size = 482, normalized size = 2.55 \begin {gather*} \frac {2 \, {\left (1575 \, b^{3} d^{5} + 280 \, {\left (2 \, a^{2} b x^{3} - a^{3} x^{2}\right )} e^{5} - 28 \, {\left (48 \, a b^{2} d x^{3} - 94 \, a^{2} b d x^{2} + 23 \, a^{3} d x\right )} e^{4} + {\left (928 \, b^{3} d^{2} x^{3} - 5168 \, a b^{2} d^{2} x^{2} + 3890 \, a^{2} b d^{2} x - 409 \, a^{3} d^{2}\right )} e^{3} + 7 \, {\left (464 \, b^{3} d^{3} x^{2} - 952 \, a b^{2} d^{3} x + 279 \, a^{2} b d^{3}\right )} e^{2} + 175 \, {\left (22 \, b^{3} d^{4} x - 17 \, a b^{2} d^{4}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{105 \, {\left (b^{4} d^{8} + a^{4} x^{4} e^{8} - 4 \, {\left (a^{3} b d x^{4} - a^{4} d x^{3}\right )} e^{7} + 2 \, {\left (3 \, a^{2} b^{2} d^{2} x^{4} - 8 \, a^{3} b d^{2} x^{3} + 3 \, a^{4} d^{2} x^{2}\right )} e^{6} - 4 \, {\left (a b^{3} d^{3} x^{4} - 6 \, a^{2} b^{2} d^{3} x^{3} + 6 \, a^{3} b d^{3} x^{2} - a^{4} d^{3} x\right )} e^{5} + {\left (b^{4} d^{4} x^{4} - 16 \, a b^{3} d^{4} x^{3} + 36 \, a^{2} b^{2} d^{4} x^{2} - 16 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} e^{4} + 4 \, {\left (b^{4} d^{5} x^{3} - 6 \, a b^{3} d^{5} x^{2} + 6 \, a^{2} b^{2} d^{5} x - a^{3} b d^{5}\right )} e^{3} + 2 \, {\left (3 \, b^{4} d^{6} x^{2} - 8 \, a b^{3} d^{6} x + 3 \, a^{2} b^{2} d^{6}\right )} e^{2} + 4 \, {\left (b^{4} d^{7} x - a b^{3} d^{7}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 524 vs.
\(2 (174) = 348\).
time = 3.45, size = 524, normalized size = 2.77 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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