Optimal. Leaf size=341 \[ -\frac {7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626,
635, 212} \begin {gather*} \frac {7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}-\frac {7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 684
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{12 d}\\ &=\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 d^2}\\ &=\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^3 d^3 e}\\ &=-\frac {7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {\left (7 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^4 d^4 e^2}\\ &=-\frac {7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {\left (7 \left (c d^2-a e^2\right )^6\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^4 d^4 e^2}\\ &=-\frac {7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.07, size = 280, normalized size = 0.82 \begin {gather*} \frac {((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (105 e^5 (a e+c d x)^5-595 c d e^4 (a e+c d x)^4 (d+e x)+1386 c^2 d^2 e^3 (a e+c d x)^3 (d+e x)^2-1686 c^3 d^3 e^2 (a e+c d x)^2 (d+e x)^3-595 c^4 d^4 e (a e+c d x) (d+e x)^4+105 c^5 d^5 (d+e x)^5\right )}{(a e+c d x) (d+e x)}+\frac {105 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 c^{9/2} d^{9/2} e^{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. \(1187\) vs.
\(2(307)=614\).
time = 0.69, size = 1188, normalized size = 3.48
method | result | size |
default | \(e^{2} \left (\frac {x \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{6 c d e}-\frac {7 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{5 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2 c d e}\right )}{12 c d e}-\frac {a \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{6 c}\right )+2 d e \left (\frac {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{5 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2 c d e}\right )+d^{2} \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )\) | \(1188\) |
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 [A]
time = 3.22, size = 1009, normalized size = 2.96 \begin {gather*} \left [\frac {{\left (105 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (70 \, c^{6} d^{10} x e^{2} - 105 \, c^{6} d^{11} e + 70 \, a^{4} c^{2} d^{2} x e^{10} - 105 \, a^{5} c d e^{11} - 7 \, {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} + 8 \, {\left (6 \, a^{2} c^{4} d^{4} x^{3} - 49 \, a^{3} c^{3} d^{4} x\right )} e^{8} + 2 \, {\left (832 \, a c^{5} d^{5} x^{4} + 156 \, a^{2} c^{4} d^{5} x^{2} - 693 \, a^{3} c^{3} d^{5}\right )} e^{7} + 20 \, {\left (64 \, c^{6} d^{6} x^{5} + 328 \, a c^{5} d^{6} x^{3} + 45 \, a^{2} c^{4} d^{6} x\right )} e^{6} + 2 \, {\left (2368 \, c^{6} d^{7} x^{4} + 4764 \, a c^{5} d^{7} x^{2} + 843 \, a^{2} c^{4} d^{7}\right )} e^{5} + 8 \, {\left (774 \, c^{6} d^{8} x^{3} + 719 \, a c^{5} d^{8} x\right )} e^{4} + {\left (3016 \, c^{6} d^{9} x^{2} + 595 \, a c^{5} d^{9}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{30720 \, c^{5} d^{5}}, -\frac {{\left (105 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (70 \, c^{6} d^{10} x e^{2} - 105 \, c^{6} d^{11} e + 70 \, a^{4} c^{2} d^{2} x e^{10} - 105 \, a^{5} c d e^{11} - 7 \, {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} + 8 \, {\left (6 \, a^{2} c^{4} d^{4} x^{3} - 49 \, a^{3} c^{3} d^{4} x\right )} e^{8} + 2 \, {\left (832 \, a c^{5} d^{5} x^{4} + 156 \, a^{2} c^{4} d^{5} x^{2} - 693 \, a^{3} c^{3} d^{5}\right )} e^{7} + 20 \, {\left (64 \, c^{6} d^{6} x^{5} + 328 \, a c^{5} d^{6} x^{3} + 45 \, a^{2} c^{4} d^{6} x\right )} e^{6} + 2 \, {\left (2368 \, c^{6} d^{7} x^{4} + 4764 \, a c^{5} d^{7} x^{2} + 843 \, a^{2} c^{4} d^{7}\right )} e^{5} + 8 \, {\left (774 \, c^{6} d^{8} x^{3} + 719 \, a c^{5} d^{8} x\right )} e^{4} + {\left (3016 \, c^{6} d^{9} x^{2} + 595 \, a c^{5} d^{9}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{15360 \, c^{5} d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 488, normalized size = 1.43 \begin {gather*} \frac {1}{7680} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d x e^{3} + \frac {{\left (37 \, c^{6} d^{7} e^{7} + 13 \, a c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (387 \, c^{6} d^{8} e^{6} + 410 \, a c^{5} d^{6} e^{8} + 3 \, a^{2} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (377 \, c^{6} d^{9} e^{5} + 1191 \, a c^{5} d^{7} e^{7} + 39 \, a^{2} c^{4} d^{5} e^{9} - 7 \, a^{3} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (35 \, c^{6} d^{10} e^{4} + 2876 \, a c^{5} d^{8} e^{6} + 450 \, a^{2} c^{4} d^{6} e^{8} - 196 \, a^{3} c^{3} d^{4} e^{10} + 35 \, a^{4} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (105 \, c^{6} d^{11} e^{3} - 595 \, a c^{5} d^{9} e^{5} - 1686 \, a^{2} c^{4} d^{7} e^{7} + 1386 \, a^{3} c^{3} d^{5} e^{9} - 595 \, a^{4} c^{2} d^{3} e^{11} + 105 \, a^{5} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} - \frac {7 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{1024 \, \sqrt {c d} c^{4} d^{4}} \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 {\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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