Optimal. Leaf size=294 \[ \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65,
214} \begin {gather*} -\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^3 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.00, size = 232, normalized size = 0.79 \begin {gather*} \frac {-\sqrt {b} \sqrt {d+e x} \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )-315 e^4 \sqrt {-b d+a e} (a+b x)^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs.
\(2(196)=392\).
time = 0.76, size = 892, normalized size = 3.03
method | result | size |
risch | \(\frac {2 e^{4} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}+\frac {\left (\frac {325 e^{5} \left (e x +d \right )^{\frac {7}{2}} a}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {325 e^{4} \left (e x +d \right )^{\frac {7}{2}} d}{64 b \left (b e x +a e \right )^{4}}+\frac {765 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{2}}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {765 e^{5} \left (e x +d \right )^{\frac {5}{2}} a d}{32 b^{2} \left (b e x +a e \right )^{4}}+\frac {765 e^{4} \left (e x +d \right )^{\frac {5}{2}} d^{2}}{64 b \left (b e x +a e \right )^{4}}+\frac {643 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{3}}{64 b^{4} \left (b e x +a e \right )^{4}}-\frac {1929 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2} d}{64 b^{3} \left (b e x +a e \right )^{4}}+\frac {1929 e^{5} \left (e x +d \right )^{\frac {3}{2}} a \,d^{2}}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {643 e^{4} \left (e x +d \right )^{\frac {3}{2}} d^{3}}{64 b \left (b e x +a e \right )^{4}}+\frac {187 e^{8} \sqrt {e x +d}\, a^{4}}{64 b^{5} \left (b e x +a e \right )^{4}}-\frac {187 e^{7} \sqrt {e x +d}\, a^{3} d}{16 b^{4} \left (b e x +a e \right )^{4}}+\frac {561 e^{6} \sqrt {e x +d}\, a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{4}}-\frac {187 e^{5} \sqrt {e x +d}\, a \,d^{3}}{16 b^{2} \left (b e x +a e \right )^{4}}+\frac {187 e^{4} \sqrt {e x +d}\, d^{4}}{64 b \left (b e x +a e \right )^{4}}-\frac {315 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a}{64 b^{5} \sqrt {b \left (a e -b d \right )}}+\frac {315 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d}{64 b^{4} \sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(531\) |
default | \(\frac {\left (315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} b d \,e^{4}-1530 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{3} d e -643 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{4} d^{3}+315 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{4} e^{4}+187 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{4} d^{4}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{4} e^{5} x^{4}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{5} d \,e^{4} x^{4}+128 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{4} e^{4} x^{4}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{3} e^{5} x^{3}+325 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {7}{2}} a \,b^{3} e -315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{5} e^{5}+1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{3} d \,e^{4} x^{2}+768 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b^{2} e^{4} x^{2}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b^{2} d \,e^{4} x +765 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{2} e^{2}-325 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {7}{2}} b^{4} d +765 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} b^{4} d^{2}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{4} d \,e^{4} x^{3}+512 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{3} e^{4} x^{3}-1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b^{2} e^{5} x^{2}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} b \,e^{5} x +643 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{3} b \,e^{3}-1929 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{2} d \,e^{2}+1929 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{3} d^{2} e +512 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} b \,e^{4} x -748 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} b d \,e^{3}+1122 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b^{2} d^{2} e^{2}-748 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{3} d^{3} e \right ) \left (b x +a \right )}{64 \sqrt {b \left (a e -b d \right )}\, b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(892\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 630, normalized size = 2.14 \begin {gather*} \left [\frac {315 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {\frac {b d - a e}{b}} e^{4} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (16 \, b^{4} d^{4} - {\left (128 \, b^{4} x^{4} + 837 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 1155 \, a^{3} b x + 315 \, a^{4}\right )} e^{4} + {\left (325 \, b^{4} d x^{3} + 555 \, a b^{3} d x^{2} + 399 \, a^{2} b^{2} d x + 105 \, a^{3} b d\right )} e^{3} + 6 \, {\left (35 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 8 \, {\left (11 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) e^{4} + {\left (16 \, b^{4} d^{4} - {\left (128 \, b^{4} x^{4} + 837 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 1155 \, a^{3} b x + 315 \, a^{4}\right )} e^{4} + {\left (325 \, b^{4} d x^{3} + 555 \, a b^{3} d x^{2} + 399 \, a^{2} b^{2} d x + 105 \, a^{3} b d\right )} e^{3} + 6 \, {\left (35 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 8 \, {\left (11 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\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 [A]
time = 1.19, size = 367, normalized size = 1.25 \begin {gather*} \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, \sqrt {x e + d} e^{4}}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {325 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {x e + d} b^{4} d^{4} e^{4} - 325 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {x e + d} a b^{3} d^{3} e^{5} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {x e + d} a^{3} b d e^{7} - 187 \, \sqrt {x e + d} a^{4} e^{8}}{64 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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