Optimal. Leaf size=366 \[ -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 686, 674,
211} \begin {gather*} \frac {3 c^5 d^5 \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 674
Rule 676
Rule 686
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {(c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx}{2 e}\\ &=-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx}{16 e^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 e^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^4 d^4\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 e^3 \left (c d^2-a e^2\right )}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 e^3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{128 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.51, size = 300, normalized size = 0.82 \begin {gather*} \frac {c^5 d^5 ((a e+c d x) (d+e x))^{5/2} \left (-\frac {\sqrt {e} \left (128 a^4 e^8+16 a^3 c d e^6 (-11 d+21 e x)+8 a^2 c^2 d^2 e^4 \left (d^2-64 d e x+31 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (5 d^3+23 d^2 e x-233 d e^2 x^2+5 e^3 x^3\right )+c^4 d^4 \left (15 d^4+70 d^3 e x+128 d^2 e^2 x^2-70 d e^3 x^3-15 e^4 x^4\right )\right )}{c^5 d^5 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 (d+e x)^5}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{5/2}}\right )}{640 e^{7/2} (d+e x)^{5/2}} \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. \(899\) vs.
\(2(322)=644\).
time = 0.76, size = 900, normalized size = 2.46
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{5} e^{5} x^{5}+75 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{6} e^{4} x^{4}+150 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{7} e^{3} x^{3}+150 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{8} e^{2} x^{2}-15 c^{4} d^{4} e^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+75 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{9} e x +10 a \,c^{3} d^{3} e^{5} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-70 c^{4} d^{5} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{10}+248 a^{2} c^{2} d^{2} e^{6} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-466 a \,c^{3} d^{4} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+128 c^{4} d^{6} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+336 a^{3} c d \,e^{7} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-512 a^{2} c^{2} d^{3} e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+46 a \,c^{3} d^{5} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+70 c^{4} d^{7} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+128 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{4} e^{8}-176 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{3} c \,d^{2} e^{6}+8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{4} e^{4}+10 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,c^{3} d^{6} e^{2}+15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{4} d^{8}\right )}{640 \left (e x +d \right )^{\frac {11}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, e^{3} \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\) | \(900\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 822 vs.
\(2 (323) = 646\).
time = 4.92, size = 1664, normalized size = 4.55 \begin {gather*} \left [-\frac {15 \, {\left (c^{5} d^{5} x^{6} e^{6} + 6 \, c^{5} d^{6} x^{5} e^{5} + 15 \, c^{5} d^{7} x^{4} e^{4} + 20 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} + 6 \, c^{5} d^{10} x e + c^{5} d^{11}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - {\left (c d x^{2} + 2 \, a d\right )} e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d^{2} e + a e^{3}} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (70 \, c^{5} d^{9} x e^{2} + 15 \, c^{5} d^{10} e - 336 \, a^{4} c d x e^{10} - 128 \, a^{5} e^{11} - 8 \, {\left (31 \, a^{3} c^{2} d^{2} x^{2} - 38 \, a^{4} c d^{2}\right )} e^{9} - 2 \, {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 424 \, a^{3} c^{2} d^{3} x\right )} e^{8} + {\left (15 \, a c^{4} d^{4} x^{4} + 714 \, a^{2} c^{3} d^{4} x^{2} - 184 \, a^{3} c^{2} d^{4}\right )} e^{7} + 2 \, {\left (40 \, a c^{4} d^{5} x^{3} - 279 \, a^{2} c^{3} d^{5} x\right )} e^{6} - {\left (15 \, c^{5} d^{6} x^{4} + 594 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e^{5} - 2 \, {\left (35 \, c^{5} d^{7} x^{3} + 12 \, a c^{4} d^{7} x\right )} e^{4} + {\left (128 \, c^{5} d^{8} x^{2} - 5 \, a c^{4} d^{8}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{1280 \, {\left (6 \, c^{3} d^{11} x e^{5} + c^{3} d^{12} e^{4} - a^{3} x^{6} e^{16} - 6 \, a^{3} d x^{5} e^{15} + 3 \, {\left (a^{2} c d^{2} x^{6} - 5 \, a^{3} d^{2} x^{4}\right )} e^{14} + 2 \, {\left (9 \, a^{2} c d^{3} x^{5} - 10 \, a^{3} d^{3} x^{3}\right )} e^{13} - 3 \, {\left (a c^{2} d^{4} x^{6} - 15 \, a^{2} c d^{4} x^{4} + 5 \, a^{3} d^{4} x^{2}\right )} e^{12} - 6 \, {\left (3 \, a c^{2} d^{5} x^{5} - 10 \, a^{2} c d^{5} x^{3} + a^{3} d^{5} x\right )} e^{11} + {\left (c^{3} d^{6} x^{6} - 45 \, a c^{2} d^{6} x^{4} + 45 \, a^{2} c d^{6} x^{2} - a^{3} d^{6}\right )} e^{10} + 6 \, {\left (c^{3} d^{7} x^{5} - 10 \, a c^{2} d^{7} x^{3} + 3 \, a^{2} c d^{7} x\right )} e^{9} + 3 \, {\left (5 \, c^{3} d^{8} x^{4} - 15 \, a c^{2} d^{8} x^{2} + a^{2} c d^{8}\right )} e^{8} + 2 \, {\left (10 \, c^{3} d^{9} x^{3} - 9 \, a c^{2} d^{9} x\right )} e^{7} + 3 \, {\left (5 \, c^{3} d^{10} x^{2} - a c^{2} d^{10}\right )} e^{6}\right )}}, -\frac {15 \, {\left (c^{5} d^{5} x^{6} e^{6} + 6 \, c^{5} d^{6} x^{5} e^{5} + 15 \, c^{5} d^{7} x^{4} e^{4} + 20 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} + 6 \, c^{5} d^{10} x e + c^{5} d^{11}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} e - a e^{3}} \sqrt {x e + d}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) + {\left (70 \, c^{5} d^{9} x e^{2} + 15 \, c^{5} d^{10} e - 336 \, a^{4} c d x e^{10} - 128 \, a^{5} e^{11} - 8 \, {\left (31 \, a^{3} c^{2} d^{2} x^{2} - 38 \, a^{4} c d^{2}\right )} e^{9} - 2 \, {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 424 \, a^{3} c^{2} d^{3} x\right )} e^{8} + {\left (15 \, a c^{4} d^{4} x^{4} + 714 \, a^{2} c^{3} d^{4} x^{2} - 184 \, a^{3} c^{2} d^{4}\right )} e^{7} + 2 \, {\left (40 \, a c^{4} d^{5} x^{3} - 279 \, a^{2} c^{3} d^{5} x\right )} e^{6} - {\left (15 \, c^{5} d^{6} x^{4} + 594 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e^{5} - 2 \, {\left (35 \, c^{5} d^{7} x^{3} + 12 \, a c^{4} d^{7} x\right )} e^{4} + {\left (128 \, c^{5} d^{8} x^{2} - 5 \, a c^{4} d^{8}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{640 \, {\left (6 \, c^{3} d^{11} x e^{5} + c^{3} d^{12} e^{4} - a^{3} x^{6} e^{16} - 6 \, a^{3} d x^{5} e^{15} + 3 \, {\left (a^{2} c d^{2} x^{6} - 5 \, a^{3} d^{2} x^{4}\right )} e^{14} + 2 \, {\left (9 \, a^{2} c d^{3} x^{5} - 10 \, a^{3} d^{3} x^{3}\right )} e^{13} - 3 \, {\left (a c^{2} d^{4} x^{6} - 15 \, a^{2} c d^{4} x^{4} + 5 \, a^{3} d^{4} x^{2}\right )} e^{12} - 6 \, {\left (3 \, a c^{2} d^{5} x^{5} - 10 \, a^{2} c d^{5} x^{3} + a^{3} d^{5} x\right )} e^{11} + {\left (c^{3} d^{6} x^{6} - 45 \, a c^{2} d^{6} x^{4} + 45 \, a^{2} c d^{6} x^{2} - a^{3} d^{6}\right )} e^{10} + 6 \, {\left (c^{3} d^{7} x^{5} - 10 \, a c^{2} d^{7} x^{3} + 3 \, a^{2} c d^{7} x\right )} e^{9} + 3 \, {\left (5 \, c^{3} d^{8} x^{4} - 15 \, a c^{2} d^{8} x^{2} + a^{2} c d^{8}\right )} e^{8} + 2 \, {\left (10 \, c^{3} d^{9} x^{3} - 9 \, a c^{2} d^{9} x\right )} e^{7} + 3 \, {\left (5 \, c^{3} d^{10} x^{2} - a c^{2} d^{10}\right )} e^{6}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs.
\(2 (323) = 646\).
time = 1.87, size = 710, normalized size = 1.94 \begin {gather*} \frac {{\left (\frac {15 \, c^{6} d^{6} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {{\left (15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{10} d^{14} e^{5} - 60 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{9} d^{12} e^{7} + 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{9} d^{12} e^{4} + 90 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{8} d^{10} e^{9} - 210 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{8} d^{10} e^{6} + 128 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{8} d^{10} e^{3} - 60 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{7} d^{8} e^{11} + 210 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{7} d^{8} e^{8} - 256 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{7} d^{8} e^{5} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{7} d^{8} e^{2} + 15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{4} c^{6} d^{6} e^{13} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c^{6} d^{6} e^{10} + 128 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c^{6} d^{6} e^{7} + 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c^{6} d^{6} e^{4} - 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c^{6} d^{6} e\right )} e^{\left (-5\right )}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{5} c^{5} d^{5}}\right )} e^{\left (-4\right )}}{640 \, c d} \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 (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{17/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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