Optimal. Leaf size=816 \[ \frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-21 a e)+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b d e-20 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 1.87, antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {756, 828, 857,
732, 435, 430} \begin {gather*} \frac {2 e \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{1155 c^3 e^3}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {c x^2+b x+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 756
Rule 828
Rule 857
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (11 c d^2-e (5 b d+a e)\right )+3 e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 c}\\ &=\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {4 \int \frac {\left (\frac {3}{4} e \left (b c^2 d^3+13 b^2 c d^2 e-58 a c^2 d^2 e-6 b^3 d e^2+27 a b c d e^2-2 a b^2 e^3+6 a^2 c e^3\right )+\frac {3}{4} e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {-\frac {3}{8} e \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )-3 e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac {\left (8 \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac {3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e^5}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (16 \sqrt {2} \sqrt {b^2-4 a c} \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac {3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3465 c^4 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+84 a c^3 d^2 e^2+13 b^3 c d e^3-84 a b c^2 d e^3-8 b^4 e^4+51 a b^2 c e^4-60 a^2 c^2 e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 32.96, size = 10848, normalized size = 13.29 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11932\) vs.
\(2(746)=1492\).
time = 1.08, size = 11933, normalized size = 14.62
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2888\) |
risch | \(\text {Expression too large to display}\) | \(4982\) |
default | \(\text {Expression too large to display}\) | \(11933\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.55, size = 869, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 3 \, {\left (11 \, b^{2} c^{4} + 36 \, a c^{5}\right )} d^{4} e^{2} + 2 \, {\left (7 \, b^{3} c^{3} - 108 \, a b c^{4}\right )} d^{3} e^{3} + 3 \, {\left (11 \, b^{4} c^{2} - 102 \, a b^{2} c^{3} + 312 \, a^{2} c^{4}\right )} d^{2} e^{4} - 6 \, {\left (8 \, b^{5} c - 69 \, a b^{3} c^{2} + 156 \, a^{2} b c^{3}\right )} d e^{5} + {\left (16 \, b^{6} - 144 \, a b^{4} c + 369 \, a^{2} b^{2} c^{2} - 180 \, a^{3} c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, {\left (b^{2} c^{4} + 6 \, a c^{5}\right )} d^{3} e^{3} + 2 \, {\left (b^{3} c^{3} - 9 \, a b c^{4}\right )} d^{2} e^{4} - {\left (5 \, b^{4} c^{2} - 36 \, a b^{2} c^{3} + 54 \, a^{2} c^{4}\right )} d e^{5} + {\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 27 \, a^{2} b c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{6} d^{4} e^{2} + {\left (105 \, c^{6} x^{4} + 140 \, b c^{5} x^{3} + 8 \, b^{4} c^{2} - 51 \, a b^{2} c^{3} + 60 \, a^{2} c^{4} + 5 \, {\left (b^{2} c^{4} + 39 \, a c^{5}\right )} x^{2} - 2 \, {\left (3 \, b^{3} c^{3} - 16 \, a b c^{4}\right )} x\right )} e^{6} + {\left (140 \, c^{6} d x^{3} + 205 \, b c^{5} d x^{2} + 2 \, {\left (7 \, b^{2} c^{4} + 163 \, a c^{5}\right )} d x - {\left (19 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} d\right )} e^{5} + {\left (5 \, c^{6} d^{2} x^{2} + 14 \, b c^{5} d^{2} x + {\left (6 \, b^{2} c^{4} + 47 \, a c^{5}\right )} d^{2}\right )} e^{4} - {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3465 \, c^{5}} \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 (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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